The Moors went behind to an Asa Hall strike but goals from skipper Darren Carter and Jamie Reckord gave them a priceless three points.
It’s Yates’ first away victory since taking charge of the club but he is fully aware that they need to put a winning run together.
Moors are still bottom of the National League table and nine points adrift of safety ahead of Monday’s home game against Maidenhead United.
“Barrow is a tough place to come and they’re on a good run at the moment but we defended brilliantly,” Yates told the club’s audioboom service.
“Barrow are the best in the division at what they do. They get the ball in the box and have terrific delivery from throw-ins and corners.
“We thoroughly deserved the win. We’ve had Maidenhead and Barrow away so it’s been tough. It’s going to be a big ask on Monday but I’ve got total belief that this group can recover and go again. We need back-to-back wins.”
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