Mbappe netted three times in just under seven minutes in Real Madrid's 4-3 win at Olympiakos on Wednesday (Thursday AEST).
"I'm very happy to have scored the goals, it's always a pleasure for me," Mbappe said.
"My teammates always set me up with good balls. I'm lucky to play for this team that has so much quality."
The France international added a fourth goal in the 60th minute after Mehdi Taremi had scored for the hosts in the 52nd minute.
Mohamed Salah's hat-trick for Liverpool at Rangers in 2022 had broken the record for the quickest ever scored in the Champions League, according to UEFA. The Egypt forward needed just six minutes and 12 seconds to score his goals after entering the match seven minutes before his opening strike.
It was Mbappe's fifth hat-trick in the Champions League and second this season. He had scored three times against Kairat in the second round of the league phase.
Lionel Messi and Cristiano Ronaldo have eight hat-tricks each in the Champions League, while Robert Lewandowski has six.
Mbappe leads the Champions League in scoring this season with nine goals, three more than Galatasaray's Victor Osimhen.
The Frenchman's goals helped Madrid end a three-game winless streak across all competitions, and moved the Spanish powerhouses to fifth place in the league-phase standings of the Champions League.
"It was important to win again," Mbappe said. "Three matches without a victory is a lot for us. We had to win to make sure we stay in the top eight, which is the goal for us now."
