The Manchester United full-back opened the scoring after 33 minutes, with club-mate Bruno Fernandes doubling Portugal's lead before Patrik Schick smashed a penalty over the bar following a VAR call at the end of the first half.
Dalot scored again shortly after the break, and Diogo Jota added gloss to the scoreline late on.
The result moved Portugal above Spain in the standings with one round of games left to play.
Antonin Barak wasted the best chance for the hosts on the half-hour mark as he headed wide, and he would rue his error when Dalot broke the deadlock from close range after Rafael Leao prevented the ball from going out of play by cutting back into the middle.
A dramatic end to the opening half saw Fernandes volley home Mario Rui's cross to double Portugal's lead, before Cristiano Ronaldo was penalised for handball at the other end, the VAR intervention resulting in a penalty that Schick wasted.
Any hopes of a comeback for the Czech Republic suffered a significant dent six minutes after the restart, Dalot netting his second of the game when he curled home a low effort from outside the penalty area.
Further opportunities were few and far between, but there was time for Portugal to extend their lead further, Jota heading home following a corner in the 82nd minute.
Milan are without Zlatan Ibrahimovic, Mario Mandzukic, Ismael Bennacer, Hakan Calhanoglu, Theo Hernandez and Ante Rebic for Thursday's first leg.
The Italian side travel to Old Trafford with just two strikers in their squad – Rafael Leao and academy product Riccardo Tonin, who has yet to feature at senior level.
However, Milan were also far from full strength for their 2-0 league win at Hellas Verona last weekend and Pioli is hoping to build confidence from that victory.
"We have a choice to make: either let ourselves be overwhelmed by the negative waves or ride the positive ones of confidence," he said.
"We will choose to ride it and I'm sure the team will put in a good performance against a difficult opponent.
"Winning our last game gives us momentum. When we play at our best we do well and against United we will be prepared.
"We have to play this type of game to our fullest ability. A positive result would give us momentum in view of the return match."
Milan are meeting United for the 11th time in a competitive match, though all 10 previous encounters have come in either the European Cup or Champions League.
The Rossoneri have not competed in Europe's flagship competition since 2013-14, while English heavyweights United exited the tournament at the group stage.
Both teams are second in their domestic leagues and United enter the match on the back of an impressive 2-0 win at Manchester City last time out.
But Pioli does not believe the winners of the two-legged affair should necessarily be considered the favourites to go all the way and lift the trophy.
"I don't think so. There are many other teams that have a chance to win it," he said. "But of course it is a prestigious match with a lot of history and past.
"We are working hard to get back to winning trophies. As [technical director Paolo] Maldini said, it takes a couple of years of continued presence in the Champions League.
"We need to tackle this path with enthusiasm and conviction."
Thursday's clash will mark a first return to Old Trafford for Diogo Dalot since joining Milan on loan last October.
The full-back struggled for appearances at United but has been used 20 times in all competitions by Pioli, including eight outings in the Europa League.
"We are happy with his performance and his growth," Pioli said. "As far as his future is concerned, we are too focused on the league and on the Europa League for that.
"Besides, it is not me who makes a decision about his future."
Milan have been eliminated on six of the last seven occasions in which they have reached the last 16 of a major European competition.
However, they have got the better of United in four of their previous five knockout ties in European competition, with no side eliminating the Red Devils more often.
United arrived at Prenton Park having lost back-to-back Premier League games against Liverpool and Burnley and the club's plight prompted some in the away end to reprise the disgruntled atmosphere that filled Old Trafford after defeat to the Clarets four days earlier.
Goals from Harry Maguire, Diogo Dalot, Jesse Lingard, Phil Jones, Anthony Martial and Mason Greenwood lightened the mood among the United faithful as Solskjaer's men breezed past League One opposition, and the manager praised his side's approach after they avoided a potential banana skin.
"If you go into the trap of playing long balls and making this game a fight, we're not used to that," Solskjaer told BT Sport. "They would have been better suited than us. We just asked our players to play and it went well.
"I think the players played football the way it should be played. It was an exceptional performance at times. It was a very difficult pitch.
"It wasn't just the Tranmere people who wanted an upset, there were a lot of others. We've just got to stay together and keep working.
"Every game you win is a good one for your confidence. Every time you score a goal and perform well, it's good.
"The attitude was spot on. That's all you can ask for. When the attitude is right, you can't fault them."
Maguire, who opened the scoring for United with a powerful 20-yard drive, and then provided the assist for Lingard to score their third in a display full of guile and attacking intent, said his first goal for the club was long overdue.
"It's been a long time coming," said Maguire. "I've had a lot of chances this year without finding the back of the net. It’s nice to help the team get into the next round.
"It was really important to get the win, coming off the back of a bad performance and result against Burnley.
"We set our stall out to put in a positive performance for the fans. We're in the next round, which is what we came to do."
