Sarr struck twice in a dramatic 2-1 aggregate win, including the decisive extra-time goal, to set up a last-eight meeting with Fiorentina.
After a goalless first leg, Palace made a fast start in Cyprus, with Sarr rounding the goalkeeper to score in the 13th minute. But despite controlling the game, they were pegged back when Enric Saborit headed home from a set-piece in the second half.
The defender was later sent off for a second yellow card after pulling down Sarr, yet Palace still struggled to break through against 10 men, with the tie going to extra time.
Sarr eventually made the difference, steering home from close range nine minutes into extra time. Late drama followed as a penalty awarded against Jean-Philippe Mateta was overturned by VAR, before Larnaca were reduced to nine men and Sarr hit the woodwork twice in search of a hat-trick.
"It's a huge achievement," said boss Oliver Glasner. "We are very pleased. In some moments we made it difficult for ourselves, especially after scoring and controlling the game, but the reaction was great.
"We had many chances to decide it earlier, but when it's only a one-goal lead, anything can happen."
Sarr's impact proved decisive once again, continuing his strong form since joining from Marseille, with this his fifth goal of the campaign.
"He's been top," said Adam Wharton. "He's scoring goals, getting assists, and he's so quick. When you get that space in behind, it works wonders having him."
Looking ahead to Fiorentina, Wharton added: "It's exciting. A European quarter-final is massive for us, the club and the fans. It's a great challenge."
Glasner also outlined plans for the break before the tie: "We are looking forward (to it) and now really make the best with these three weeks because our next competitive game is against Fiorentina at Selhurst Park.
"We will have a few days off here, and many of our players go to their national team, and then we will prepare, we will make a friendly to stay in the rhythm, and then it's good.
"I expect Eddie Nketiah back so everybody will help us because then it will be a very tight schedule until the end of the season, and hopefully our European journey continues."
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