The visitors came close late on through substitute Dwight Gayle but didn't warrant enough threat during the 90 minutes in what was a tight affair.
Quiet first half
In the battle against the wind, it was the Magpies who threatened first when Isaac Hayden delivered an inviting cross for Andy Carroll to latch onto, but his header was wide.
It didn't take long for the hosts to grow into the game though, with Dwight McNeil forcing a sharp save from Martin Dubravka.
VAR became involved on 13 minutes in response to claims of a handball from Paul Dummett, but the home side had to settle for just a corner.
It was Hayden again who tried to make something happen for United when he picked the ball up on 25 minutes and tried his luck from distance, but the ball never troubled Nick Pope in goal.
Wood on the spot
After a rather unentertaining first half, finally, Burnley had something to cheer about when a McNeil corner found Wood, and the New Zealander nodded past Dubravka.
After the game Steve Bruce was adamant there was a foul in the build-up, but referee Tim Robinson didn't agree, in what was his first Premier League encounter.
As time ticked on Newcastle huffed and puffed and Joelinton headed a Carroll cross wide on 64 minutes, which was followed by an important stop from Dubravka to deny Jack Cork three minutes later.
United's injuries proving costly
With the absence of Allan Saint-Maximin, Miguel Almiron and Jonjo Shelvey, this was where Bruce needed to rely on those who have done more spectating than playing this season.
In the last ten minutes, they had four forwards on the pitch after Yoshinori Muto and Gayle joined the action.
United had a great opportunity to level in the closing stages when Gayle met a Joelinton cross but he could only divert wide.
Burnley climb to twelve and now sit a point behind the Magpies, as the bottom half grows even tighter after Bournemouth's win at Chelsea.