Danis, the Islanders' third-string goalie, could be in line for more action with regular backup Joey MacDonald because No. 1 netminder Rick DiPietro will be out four-to-six weeks following knee surgery on Friday.
The Canadiens got two goals and two assists from Tomas Plekanec and one from Long Island native Christopher Higgins. Carey Price made 27 saves.
Former Canadiens defenseman Mark Streit scored for the Islanders, along with Jon Sim, Bill Guerin and Frans Nielsen.
The Islanders are winless in six games (0-5-1) since a 4-3 overtime victory at Tampa Bay on Oct. 16. Danis, who spent time in the Canadiens organization, stopped 27 shots.
Montreal took a 1-0 lead 2:03 in when Kovalev took Plekanec's pass in the low slot and beat Danis with a quick wrist shot.
New York tied it at 3:47 on its first power play. Streit ripped a right point slap shot that beat a screened Price.
The Islanders took their first lead after a span of 317 minutes, 4 seconds of playing time over five games when Sim sent a slap shot from the top of the left circle between Price's pads at 12:27 that made it 2-1. New York hadn't been ahead in a game since Oct. 16 when the Islanders beat Tampa Bay in overtime.
New York capitalized on another power play early in the second period to grab a 3-1 lead when Guerin scored on a tip-in.
Nielsen made it 4-1 on an in-close wrist shot that hit the top of the net at 16:43.
Plekanec scored twice in a 2:07 span to close the gap to 4-3 in the third period. His first goal came at 7:48 on a rebound, and his second, at 9:55 was scored with a quick wrist shot.
Higgins tied it at 14:06.
Notes: The Islanders debuted their third jerseys, which look like the first uniform the team wore back in the early 1970s. One difference is that the stick in the logo has four stripes instead of the usual three, to signify each of New York's Stanley Cup championships. ... Price had allowed only 13 goals in his first seven games for a goals-against average of 1.91.