The Pro14 side edged to a 15-12 victory in Bilbao, with a lively San Mames crowd treated to an absorbing match.
Despite the absence of any tries, there was no lack of drama and, after Sexton had himself kicked three penalties, Isa Nacewa added two late on as Leinster took the lead for the first time with a couple of minutes to spare.
A relieved and delighted Sexton, who has been with the club for all of its triumphs in Europe's premier competition, said: "We did it the hard way. We couldn't get going in the game. Every time we got into their 22 the ball was killed.
"It was very greasy and they slowed it down at the ruck, but I can't believe we are champions again.
"I'm so happy for the young lads and the coaches."
Asked about the influence of Stuart Lancaster, who departed his role as England coach under a cloud after the nation's abysmal 2015 Rugby World Cup campaign, the Ireland star added: "Stuart Lancaster has been brilliant since he has come in.
"We had a near miss last year which we learnt a lot from, but the belief he has instilled in us has meant a lot."
Sexton is now hoping to finish the season on a high, with a domestic semi-final against Munster next on the agenda.
"Two more wins in the league before we go to Australia will cap off the season," the 32-year-old said. "It will be a tough game against Munster who will come out hard at us now we are European champions."