The 2020 Mexican Guardianes runners-up, who were reduced to 10 men after Arturo Ortiz's 63rd-minute red card, will face the winner of the tie between New York City and Seattle Sounders.
Seattle won the first leg at home 3-1, with the second leg to be played on Wednesday, ahead of the continental decider with the first leg scheduled for later this month.
Juan Dinneno's double in the first leg gave Pumas the advantage in both scoreline and complexion, in comparison to what was a cagey first leg.
With Cruz Azul needing to chase the game and tie, space was not a premium on Tuesday and it was largely characterised by end-to-end football, with chances for both teams.
Uriel Antuna came closest for the hosts, but they were lacking composure in both the final pass and in front of goal.
Cruz Azul were given a small lifeline just past the hour, with Ortiz sent off for dragging down Santiago Gimenez as the last man in the 63rd minute.
Pumas were even denied a penalty in the 75th minute and managed to hold on. They will face the winner of the tie between New York City and the Seattle Sounders.
Alves was arrested in Barcelona on Friday after it was alleged he sexually assaulted a woman in a nightclub bathroom.
The 39-year-old, who became the oldest player to feature in a World Cup match for Brazil when he played against Cameroon in Qatar last month, has denied assaulting the woman.
According to widespread reports, Alves – who has also played for Paris Saint-Germain and Sevilla – gave a statement to the Mossos d'Esquadra police force before the public prosecutor's office requested that he should be detained and not granted bail.
Spanish news agency EFE reported how that application, plus a private prosecution brought by the woman who made the allegation, would be put to the investigating judge to determine whether Alves is kept in prison while investigations continue.
Subsequently, Pumas confirmed on Friday, via a short statement on their website and a media briefing from club president Leopoldo Silva, that they had terminated Alves' contract.
Alves joined Mexican side Pumas in 2022 after his second spell at Barca came to an end.
The Sounders beat Pumas UNAM 3-0 at Lumen Field on Wednesday to complete a 5-2 aggregate triumph and finally deliver a first Champions League title for MLS.
Seattle are the league's third CONCACAF champions but first in the Champions League era after a series of heartbreaks for rival clubs.
Two goals from Raul Ruidiaz and a late clincher from captain Nicolas Lodeiro sparked scenes of mass celebration in front of a raucous, record crowd of over 68,741.
Garth Lagerwey, the Sounders' general manager, gave Extratime his assessment: "Immortality. You get into sports for stuff people can never take away.
"This will be written down, it will be there forever. Hopefully it's the first of many."
Real Salt Lake, CF Montreal, Toronto FC and Los Angeles FC had each previously fallen at this final hurdle.
Now, for MLS, Lagerwey says, Seattle are "the symbol, we're the tip of the spear, we pushed through, we finally did it, we vanquished the demons".
"But everybody's welcome," he added. "We want a crowded mountaintop up here. We don't want to be up here by ourselves."
Playing just hours after Madrid had completed an epic semi-final fightback against Manchester City in UEFA's Champions League, Lagerwey could not help consider a Club World Cup clash with a European giant.
"We're going to play against Real Madrid or Liverpool in a real game for a trophy," he said, with Seattle's place in the tournament secure. "I feel like a little kid. This is the stuff you dream of."
The Sounders had already won two MLS Cups, a Supporters' Shield and four U.S. Open Cups, but this victory takes the club to another level entirely.
"I think we're going to become a global club now," the GM added.
"I've got to think my phone's got to start ringing once some people see what our fanbase, our building... it's as good a soccer environment as anywhere in the world. It just is. This is a pretty special place."
The Blaugrana ran riot in their traditional Camp Nou curtain-raiser, with goals from Pedri (two), Lewandowski, Ousmane Dembele, Pierre-Emerick Aubameyang and Frenkie de Jong condemning the Mexican side to a crushing loss.
Barca took just 20 minutes to establish a 4-0 lead as Pedri scored twice after laying on new signing Lewandowski's opener, with the Poland striker returning the favour to tee up the skilful midfielder's second strike.
Lewandowski – who scored 50 goals in all competitions during his final season with Bundesliga champions Bayern Munich last term – was named as man of the match, leading Pedri to shower him with praise.
"It was important for us to win in front of our fans and we have achieved it by playing a good game," the midfielder said.
"The coach asks us all to reach the top and that we score goals, and I hope it continues like this. We've had a good pre-season and we wanted to start scoring soon.
"It is a luxury to play with Lewandowski, I really enjoy seeing how he finishes and how he links up.
"The president [Joan Laporta] and everyone have done a great job, and we will fight until May for it to be a very beautiful year for us all."
With Lewandowski joining Franck Kessie, Andreas Christensen, Jules Kounde and Raphinha in moving to Camp Nou ahead of the 2022-23 season, hopes are high that Barca can challenge for silverware – even if speculation persists that they may be unable to register their newcomers.
For Xavi, the performance against Pumas – for whom Barca legend Dani Alves featured – represented the standard the Catalan giants must hit in every match.
"Everything went well tonight, I'm very happy with the good game and for the fans, who have been excited. This is the way," Xavi declared.
"We are in a good dynamic of play and results, but we have to continue working for this to work. The sensations are good, yes, but there is no need to lower the pace or the intensity during the 90 minutes.
"Today it is understood that we relaxed a bit because we were 4-0 very early and it was a friendly, but in official competition you cannot lower your guard."
Barca begin their LaLiga campaign at home to Rayo Vallecano next Saturday.
Trailing to a double from Juan Dinenno, Seattle battled back in Mexico City to make it honours even ahead of next Wednesday's second leg at Lumen Field. Lodeiro's leveller came in the ninth minute of stoppage time.
This opener at Estadio Olímpico Universitario saw Dinenno take the lead with a twice-taken penalty in the 38th minute. His first attempt was saved by Stefan Frei, but the goalkeeper was ruled to have stepped off his line.
Dinenno headed in powerfully for Pumas' second goal in the 48th minute, but the comeback began when Seattle were awarded a penalty for a handball by Sebastian Saucedo, and Lodeiro drove home in the 77th minute.
Lodeiro fired into the right corner on that occasion and was handed another spot-kick deep into stoppage time.
Cristian Roldan was judged to have been fouled on the right side of the penalty area by veteran defender Efrain Velarde, and Lodeiro chose to go the same way, again beating goalkeeper Alfredo Talavera with a powerful strike.
A Juan Dinenno first-half brace ultimately separated the two side in what was an intense and tight contest.
Cruz Azul were denied the lead in the 20th minute after well-worked move, with Uriel Antuna ruled offside for his finish from Ignacio Rivero's deft reverse pass.
Pumas somehow did not break the deadlock after Cruz Azul goalkeeper Sebastian Jurado's charge for a cross, as Luis Abram scrambled to clear Favio Alvarez's shot off the line.
Pumas eventually got off the mark through Dinenno in the 37th minute and in similarly chaotic circumstances, with another Abram clearance off the line deflecting off the post, then onto Dinenno's foot and in.
He doubled the margin for Pumas in a more conventional manner in first-half injury time, reading Alan Mozo's cross best to powerfully head home past Jurado.
Cruz Azul gave themselves a late lifeline in the tie, though, with Christian Tabo's goal in the 83rd minute.
Alves played alongside Messi for eight years during his first spell at Camp Nou, forming a devastating relationship with the Argentinian on the right flank.
Both players were pillars of a Barcelona team that enjoyed a prolonged period of dominance both in Spain and Europe, with Alves winning 23 trophies during his initial eight-season stint in Catalonia.
And yet, despite seeing Messi's brilliance up close and personal, Alves – who now plays in Mexico for UNAM Pumas – still believes Pele's significance is at an altogether different level.
Speaking to Mexico, Real Madrid and Pumas legend Hugo Sanchez on ESPN, Alves was asked to pick between Messi and Pele – he said: "I choose Pele for one reason: he changed football. Messi changed a generation.
"All those people who are more than football, I'm going to choose them. And besides, I'm Brazilian.
"For me, Pele should not be part of this ['GOAT'] debate. If you ask me, who is the best player in history? I would say Messi. Pele doesn't come into it."
Cristiano Ronaldo was not brought into the 'GOAT' discussion, though Alves did appraise his former Clasico rival when asked about his toughest opponent.
"Cristiano Ronaldo," Alves added. "This b****** doesn't let you breathe for a second. I didn't do too badly, but it's difficult. He's a scoring machine."
Now 39, Alves is continuing his illustrious and immensely successful career in Liga MX having signed a one-year deal with Pumas in July.
He made the switch to Mexico City after spending eight months back at Barcelona, returning to help out his former team-mate Xavi.
It remains to be seen if he will be at Pumas beyond the current season, though he suggests thoughts of retirement have not firmly taken root, even joking he could play for another 10 years if Messi breaks his world record of 47 trophies won.
"First, before requesting anything [another season with Pumas], I like to deliver," he said.
"I have a year to prove myself with [Pumas]. I wouldn't have a problem [staying another year]. I need to show them that I can, that together we can build a very cool team in Mexico, that plays in a different way, that [lets] young people take a step forward.
"I don't like to think about the future because it's always unknown. I let life take me. If Messi passes me [in trophies] then I'm going to play until I'm 50!"
With the scores level to open the game, it proved to be a very cagey opening 45 minutes and the Sounders faced early difficulty, with Nouhou Tolo and Joao Paulo forced off with injuries.
Brian Schmetzer's side got a break just before the interval, however, with Raul Ruidiaz scoring following a scramble in the penalty area in the 45th minute.
Needing to chase the game with the ball, Pumas really struggled to create anything of substance in the second half as the Sounders absorbed pressure.
Indicative of the second leg's complexion, the best chance with the tie in the balance fell to Seattle via Alex Roldan, who intercepted a rushed throw out from Alejandro Talavera in the Pumas goal.
Ruidiaz eventually wrapped up the tie and trophy for Seattle, scoring in transition in the 80th minute. After his two penalties in the first leg, Nico Lodeiro scored Seattle's third in the 87th minute after another Pumas loss of possession.
With the scores level to open the game, it proved to be a very cagey opening 45 minutes and the Sounders faced early difficulty, with Nouhou Tolo and Joao Paulo forced off with injuries.
Brian Schmetzer's side got a break just before the interval, however, with Raul Ruidiaz scoring following a scramble in the penalty area in the 45th minute.
Needing to chase the game with the ball, Pumas really struggled to create anything of substance in the second half as the Sounders absorbed pressure.
Indicative of the second leg's complexion, the best chance with the tie in the balance fell to Seattle via Alex Roldan, who intercepted a rushed throw out from Alejandro Talavera in the Pumas goal.
Ruidiaz eventually wrapped up the tie and trophy for Seattle, scoring in transition in the 80th minute. After his two penalties in the first leg, Nico Lodeiro scored Seattle's third in the 87th minute after another Pumas loss of possession.
Brazilian right-back Alves was arrested in Barcelona on Friday after it was alleged he assaulted a woman in a nightclub bathroom.
The 39-year-old denied committing the offence when he spoke to Spanish broadcaster Antena 3 in early January.
Alves, who has also played for Paris Saint-Germain and Sevilla, gave a statement to the Mossos d'Esquadra police force on Friday before the public prosecutor's office requested that he should be detained and not granted bail.
He has been kept in prison while investigations continue, and his contract with Mexican club Pumas UNAM has been terminated.
Xavi, a former Barcelona team-mate of Alves, signed the defender for a short second Barcelona stint last season.
Speaking in a Barcelona pre-match press conference on Saturday, Xavi said: "It is difficult to comment on a situation like this. I am surprised, shocked, in a state of shock.
"It is a matter of justice. Justice will dictate whatever."
Alves became the oldest player to feature in a World Cup match for Brazil when he played against Cameroon in Qatar last month.
