Connacht's danger man, who has yet to feature for Ireland at Test level, caught Shaun Berne, Girvan Dempsey and Kearney all off guard as he cut in from the left and sprinted clear and under the posts.
New Zealander Miah Nikora added the conversion and although Connacht conceded a raft of penalties in the opening quarter, Leinster's sole reward was a fifth minute strike from Sexton.
The hosts' territorial dominance came to nothing and they were rocked again when Dempsey was robbed at a ruck and Carr linked with Nikora before gleefully dotting down to the left of the posts.
Nikora, with a considerable wind behind him, fired over a brilliant conversion to increase the pressure on Leinster who were missing the likes of Brian O'Driscoll, Gordon D'Arcy, Cian Healy and Jamie Heaslip.
Michael Cheika's men looked back on track, six minutes later, as Kearney dummied his way past Brian Tuohy and Aidan Wynne, just outside the visitors' 22, on the way to a well-taken try.
Sexton failed to convert and also pushed an injury-time penalty effort to the right of the posts, as Connacht took a six-point advantage into the break.
With Leo Cullen and his forward colleagues getting more of a grip on matters up front, Leinster reduced the arrears to three points thanks to Sexton - although the recent Six Nations debutant's radar was off in the 50th minute when he had a chance to level.
Drama followed as, with twelve minutes to go, Connacht lost full-back and captain for the day, Gavin Duffy, to the sin-bin for a punch, spotted by the touch judge.
The television replay showed Duffy tangling with Fergus McFadden off-the-ball, but there was little or no evidence of a punch.
With the electric Carr continuing to threaten, Connacht looked like they might survive and a rousing breakaway almost had the perfect ending as Cronin sped towards the line.
However, a desperate tackle from Dempsey felled the hooker, and Kearney got his arms under the ball to prevent a try.
Keatley then missed the easier of his two drop goal shots, from in front of the posts, and although Leinster scrum half Eoin Reddan did likewise at the other end, his half-back partner Sexton kept his nerve to coolly slot a 40-metre drop and dash Connacht's hopes of a famous derby win.
Speaking afterwards, a clearly frustrated Michael Cheika said: "Over the balance of play I think Connacht probably deserved to win. They dominated the contact area but I was pleased that we showed good character to get into a position where we could win the game.
"It's going to make for some hard viewing and if we think we can play like that again we have another thing coming. They dominated the contact area and it's like we lost in there in the dressing room."
|Leinster Rugby Score Card|
|Connacht Rugby Score Card|