David Ajiboye scored the only goal of the match in the 16th minute as the Yellow Army moved back into a play-off place in the EFL League Two table.
Story of the match
Sutton could have been in front after four minutes as Ajiboye drove down the right byline and sent in a cross, but Will Randall could only direct the ball wide of the target.
Soon after, Randall had another opportunity as he collected the ball in space on the edge of the penalty area, but he fired over.
Two minutes later, the Yellow Army were in front. Ajiboye cut inside past two Bluebirds and he beat Paul Farman with a low, curling drive.
Zanzala should have leveled moments later as he cleverly controlled a ball over the top, but his effort was wide of Bouzanis' goal.
Just before the half, Barrow again could have equalized. From a corner, Tom White had a go from 20 yards out, but he went just wide of the post.
After Kenny Davis fired over, Sutton were on the defense as Robbie Gotts fed Josh Gordon, who cut inside on his left foot, but Bouzanis was able to collect easily.
Ajiboye nearly doubled his and the hosts' tally as he collected the ball on the edge of the box, but Farman shifted to his right to gather.
The Barrow 'keeper was called into action again when he collected Omar Bugiel's 25-yard shot.
Zanzala set up Kay on the left side of the box, but his low drive took a deflection and harmlessly fell into the side netting.
Substitute Mark Ellis headed just wide before Gotts saw his powerful drive superbly tipped over by Bouzanis.
Farman then rescued his defence, who gave Donovan Wilson far too much space, but the Bluebirds shot-stopper pawed the ball away from the striker's feet.
That was the last meaningful action of the match as Sutton continue to impress in their debut season in the Football League.
Man of the match: David Ajiboye
He grabbed the match-winner and had other chances to extend Sutton's lead. He's the latest to contribute to arguably the most surprising team in the Football League this season.