The Rangers were 2:39 away from falling to 0-4 for the first time since 1998-99 but defenseman Brendan Smith tied the game, assuring the Rangers of at least a point.
Skjei ensured New York would get two points when he slipped a 13-foot wrist shot from near the left circle over Aaron Dell's shoulder, giving new coach David Quinn his first NHL win. It was Skjei's second career game-winning goal.
Skjei scored after Sharks defenseman Erik Karlsson committed a giveaway deep in his offensive zone when his pass from behind the net, intended for Tomas Hertl, was intercepted by the Rangers.
Before Smith tied the game and Skjei won it, rookie Brett Howden scored in the first period.
Henrik Lundqvist recorded his 432nd career win by stopping 41 shots. It was his first game this year where he faced at least 40 shots and he made 31 of those stops in the first 40 minutes.
Burns recorded his first multi-point game of the season by setting up goals by Marcus Sorenson in the first period and Joonas Donskoi in the second.
Dell made 24 saves in his second straight start.
The Sharks took the lead 84 seconds into New York's power play when Burns stripped Kevin Hayes near the blue line in the offensive zone. Sorensen wound up with the puck on a breakaway, made a move around Skjei and fired a rising wrist shot over Lundqvist's left shoulder with 15:29 left.
New York tied the game over nine minutes later thanks to Howden's deft manoeuvre in the low slot.
A pass from Chris Krieder went behind the net to Mats Zuccarello, who emerged from behind the net and slid the puck to Howden. Howden kept possession while being draped by Burns and slid a backhander through Burns' legs and by Dell.
The Sharks took a 2-1 lead 4:59 into the second when Donskoi finished off a 3-on-1 with Antii Suomela and the lead held until Smith tied it.
Smith scored the tying goal from the slot as he eluded Evander Kane and sent Pavel Buchnevich's backhanded pass from over the net flying over Dell's shoulder.
(With inputs from agencies.)