Top seed Alcaraz only needed an hour and nine minutes to beat Russian Khachanov 6-0 6-2 in a clay-court masterclass on Friday.
The 19-year-old Spaniard moved a step closer to winning a fifth title this year with a dominant performance, breaking the Russian five times in a one-sided contest.
Alcaraz hit 21 winners as Khachanov was given a harsh lesson and the teenager will next face Alex Molcan, who was 2-0 up in the second set after winning the first on a tie-break when Borna Coric retired due to injury.
"I played unbelievable today, probably one of my best matches this year," Alcaraz said of his win against Khachanov. "I'm pretty happy with the performance that I'm playing in this tournament. Yesterday [a victory over Filip Krajinovic] was amazing as well.
"I'm training every day to be solid and at the same time to be aggressive. That is my game."
Francisco Cerundolo and Lorenzo Musetti will contest the other semi-final in Hamburg after beating Aslan Karatsev and Alejandro Davidovich Fokina respectively.
A rejuvenated Thiem is enjoying another hugely encouraging week and will take on Berrettini in his first semi-final of the year following a 6-4 6-3 defeat of Peruvian qualifier Juan Pablo Varillas in Gstaad.
Austrian Thiem has not played in a final since winning his only grand slam at the 2020 US Open, but he claimed his first ATP-level victory for 14 months in Bastad last week and is through to his first semi-final since May 2021 after bossing the baseline battle with Varillas.
It was not all plain sailing for Berrettini, who roared back to beat Pedro Martinez 3-6 7-6 (7-5) 6-1 after the Spaniard was in command of a second-set tie-break at 5-1, but the second seed extended his winning run to 11 matches.
Top seed Casper Ruud got the better of Jaume Munar 7-6 (7-3) 7-6 (7-4), booking a showdown with Albert Ramos-Vinolas, who came through a three-set battle with Nicolas Jarry.
Gaston, ranked 155th in the world, booked his place in his first tour-level final with a 3-6 6-3 6-3 victory over Laslo Djere of Serbia on Saturday.
The 20-year-old had never advanced past the second round at an ATP Tour event before this week. He has now won three matches in a row in a third set.
"I tried to take more time with my game," Gaston said. "I made a lot of mistakes in the first set, so I tried to change my game. My serve was better in the second and third set.
"I am really happy to be in the final. It is going to be a good match, but I will try to enjoy [this] moment."
Ruud, meanwhile, enjoyed a rather more straightforward procession in his semi-final, with world number 249 Vit Kopriva little match for the 22-year-old Norwegian.
The world number 14 needed just 74 minutes to win 6-3 6-0, reaching his second final in as many weeks following his success in Bastad.
Ruud has won 23 clay-court matches already in 2021 and is in the hunt for his fourth career title.
Carreno Busta fell to a 6-3 1-6 7-6 (7-5) second-round defeat against Slovakian Alex Molcan, who progressed to a fourth ATP Tour quarter-final this season.
World number 48 Molcan will next meet Borna Coric after the Croatian eased past Tallon Griekspoor 6-3 6-4, registering back-to-back tour-levels wins for the first time in 17 months.
Second seed Rublev coasted through his first-round clash with Ricardas Berankis, triumphing 6-3 6-4 as he improved his record at the ATP 500 clay-court event in Germany to 10-2.
"I am happy with my performance. Ricardas is a really tough player," said world number eight Rublev, who is aiming for his fourth title of the season after success in Marseille, Dubai and Belgrade.
"He hits the ball really hard and plays fast, so I had to bring intensity from the first game. I am really happy with how I was serving today and I hit some great shots from the baseline.
"The game he broke me in the second set he played really well. He returned really aggressively and I don’t think I made one first serve. But then I kept [my] focus and broke him straight away again."
Alejandro Davidovich Fokina was another comfortable winner, easing past Jozef Kovalik 6-2 6-2 to set up a quarter-final contest with Lorenzo Musetti, who beat Emil Ruusuvuori 6-4 7-5.
Play was suspended early at the Swiss Open due to bad weather, with Spaniard Jaume Munar downing Alexander Ritschard 6-3 6-3 in the only match of the day to reach the last eight.
That meant top seed Casper Ruud will have to wait until Thursday to face Jiri Lehecka in Gstaad, with third favourite Roberto Bautista Agut also seeing his match with Juan Pablo Varillas pushed back a day.
Garin, the fourth seed, responded impressively to going a set down and then got to match point four times in their third-set tie-break, but he could not hold his nerve and Gaston got the job done 6-4 1-6 7-6 (13-11).
The Chilean was on the back foot right from the start Gaston broke him in the first game, and although Garin did hit back to make it 3-3, his opponent quickly had the advantage again and went on to seal the first set.
Garin's response was emphatic and clinical, dropping just three points on his serve and taking both of the break points that came his way to level the match.
The pair could not be separated in a back-and-forth third set, but Garin was the first to get match point at 6-5 in the tie-break – they would trade another six equally between them before Gaston finally prevailed at 13-11, leaving the 20-year-old satisfied.
"I am very happy to be in my first [ATP Tour] semi-final," Gaston said after.
"It was a great fight. It was a really long match. I always tried to play my game, and it is a good win for me. I hope I can continue like that."
Up next for Gaston will be Laslo Djere after he saw off Arthur Rinderknech 6-4 6-7 (5-7) 6-4, while Casper Ruud – the highest seed remaining – is also into the final four.
The Norwegian was victorious at the Swedish Open last week, his second title of the year, and dispatched Benoit Paire on Friday to reach another semi.
Ruud won 6-2 5-7 6-3 in just under two hours and will face Vit Kopriva, a qualifier, who secured his first semi-final appearance thanks to a stunning 6-1 6-0 win over Mikael Ymer.
At the Croatia Open in Umag, top seed Albert Ramos-Vinolas cruised into the last four with a straightforward 6-2 6-1 defeat of Stefano Travaglia, setting up a meeting with either Filip Krajinovic or Carlos Alcaraz.
The other semi will be contested between Richard Gasquet and Daniel Altmaier, who produced something of a shock by ousting second seed Dusan Lajovic 6-2 6-4.
The veteran Frenchman came through against Damir Dzumhur 6-3 7-6 (9-7), clinching the second set at the fourth opportunity.
Second seed Rublev was the latest scalp of Cerundolo's fine run of form, the Argentine making it seven consecutive wins by wrapping up a 6-4 6-2 success in just an hour and a half in Germany.
Having furthered his momentum with a second top-10 win in as many weeks, Cerundolo said: "It is the first time it has happened for me. I am playing against the top guys and I am playing well, it is unbelievable."
Cerundolo will face Aslan Karatsev in the last eight after he recovered from a set down to beat Daniel Elahi Galan 3-6 6-3 6-4.
World number six Alcaraz fared better than Rublev, registering a 7-6 (7-4) 6-3 win over Krajinovic to tee up a final-four meeting with Karen Khachanov, who beat Fabio Fognini in straight sets.
Meanwhile, a host of big names including Casper Rudd and Matteo Berrettini reached the last eight of the Swiss Open in Gstaad, as Dominic Thiem continued his revival with a win over Federico Delbonis.
Thiem reached his second quarter-final in as many weeks with an impressive 7-6 (8-6) 6-3 triumph despite letting a 5-2 lead slip in the opener, registering his first tour-level straight-sets victory since May 2021.
Having set up a meeting with Juan Pablo Varillas, the 2020 US Open champion beamed: "I have won two matches which means a lot to me right now, I really need those ranking points. Every match win is something very special."
The Austrian will be joined in the next round by first and second seeds Ruud and Berrettini, who eased past Jiri Lehecka and Richard Gasquet respectively.
Spanish duo Albert Ramos-Vinolas and Pedro Martinez also both progressed to the last eight despite dropping a set, with the latter setting up an enticing clash with Berrettini on Friday.
Ruud won the Nordea Open last weekend and added a fourth ATP Tour title of his career with a 6-3 6-2 defeat of Frenchman Gaston.
The Norwegian took his tally of tournament victories on clay this year to three, having also won the Geneva Open in May, and the 22-year-old will rise to ninth spot in the ATP Race To Turin on Monday.
Ruud saved nine of the 10 break points earned by first-time ATP Tour finalist Gaston, who only won 51 per cent of points behind his first serve.
Gaston, who had never been beyond the second round at an ATP Tour event before this week, was broken twice in each set as the 20-year-old was unable to halt Ruud's brilliant run on clay.
Left-hander Gaston broke straight back after Ruud went 3-1 up in the opening set but failed to back that up with a hold.
Ruud had as many as 16 break-point opportunities but got the job done in an hour and 34 minutes.
The world number 10, who has recently been praised by both Roger Federer and Rafael Nadal, took the first set 6-2 with ease before Kopriva, making his ATP Tour debut this week, emphatically fought back 6-3 6-2 to secure a memorable comeback victory.
Number three seed Casper Ruud battled past Dennis Novak 6-4 7-6 (7-5), while Benoit Paire, who only had two ATP wins in 2021 before the Hamburg European Open last week, was 6-4 to the good before Tallon Griekspoor was forced to retire due to injury.
Ruud, who has already won two ATP events in 2021 and three in his career, will now meet sixth seed Paire in the quarter-final stage.
But eighth seed Feliciano Lopez did not enjoy similar success as the world number 96 Mikael Ymer fought past the Spaniard to win 6-7 (6-8) 7-6 (9-7) 6-4 and set-up a quarter-final clash with Kopriva.
At the Umag Open in Croatia, third seed Filip Krajinovic breezed past Radu Albot to record a 6-4 6-2 win.
Krajinovic's countryman and number two ranked player Dusan Lajovic also went through with a 7-5 6-4 win against Bernabe Zapata Miralles.
