Evans also added 10 rebounds and seven assists, while Anthony Davis tallied 24 points and 14 boards for the Pelicans, winners in three straight and four of five. Brian Roberts scored 17 points and Anthony Morrow and Austin Rivers had 11 and 10 points, respectively.
Paul Pierce led the way with 24 points for Brooklyn, which had its four-game winning streak come to an end. Deron Williams scored 23 and reigning Eastern Conference Player of the Week Joe Johnson had 15.
Brooklyn led 66-44 early in the third quarter after a pair of Johnson free throws when New Orleans made its charge. The lead trickled into single digits midway through the frame and after Evans' four-point play ended the third, New Orleans was within three points at 76-73.
The Nets pushed it back to 84-75 after Mirza Teletovic's three-point play at 9:37, but New Orleans chipped away at its deficit over the next few minutes. Evans tied it at 89-all with his reverse layup with 4:52 to play, and the teams played to a draw the rest of the way.
The Pelicans were up five on a set of Evans foul shots with just over a minute left in regulation, but Mason Plumlee flushed home a rebound and Pierce hit a triple from the left wing to tie it. New Orleans came up small on its final look, as Morrow's contested jumper at the buzzer was off the mark.
Roberts and Morrow hit back-to-back triples midway through the extra session, giving the hosts a 106-100 lead. After Johnson dribbled the ball off his legs and out-of-bounds for a costly turnover, Morrow sealed it with another long ball from the top of the key.
Earlier, Pierce and Shaun Livingston each had eight in the first quarter to help the visitors to a 26-20 lead. It was 58-42 at halftime.