Barron's tally which went in off his skate after defenceman Neal Pionk shot it from the right point and hit captain Adam Lowry's stick, crossed the line late in the first period.
The goal was Barron's 9th of the season.
Connor Hellebuyck made 18 saves for Winnipeg, to earn his 3rd shutout of the season.
The win was Winnipeg's second straight by a one goal margin, after besting the Pittsburgh Penguins Saturday night.
The offensively challenged Jets haven't scored more than two goals for their last eight straight games.
Kaapo Kahkonen stopped 39 shots for San Jose, who dropped to 14-33-5 on the season.
The Sharks have lost two straight.
Morgan Barron gave the Jets the lead 17:43 into the first when the puck went in off his skate off a point shot from Neal Pionk which was also redirected by Adam Lowry.
The goal was his 9th of the season.
There was no scoring in the second and third periods.
SHOTS-SAN JOSE 18 WINNIPEG 40
ATTENDANCE-13,786
