The Magpies, now in the hands of former Clarets boss Eddie Howe, collected their first success of the season, as well as their first Premier League clean sheet of the campaign.
The breakthrough came five minutes before the break when Nick Pope spilled Joe Willock's delivery following a collision with Fabian Schar.
Callum Wilson was the grateful recipient of the mishap as the striker lifted the ball into the roof of the net, despite the best efforts of Chris Wood on the line.
The hosts, who lost Maxwel Cornet to injury just before Wilson's winner, had earlier hit the post through Johann Berg Gudmundsson.
And substitute Jay Rodriguez had a late finish ruled out as the linesman's flag was raised.
1. Nick Pope 6
Seemed to be in control of the ball in the lead up to Wilson's winner, until colliding with Schar. Made an impeccable fingertip stop to turn Almiron's effort wide in the second half before pushing Shelvey's attempt over the bar.
2. Matt Lowton 6
Limited the threat of Saint-Maximin, keeping the Frenchman at arm's length for large spells of the game. Looked assured on the ball, distribution was generally precise, but sacrificed late on as Roberts was introduced from the bench.
3. Charlie Taylor 7
Showed superb awareness to get goal side of Wilson from Saint-Maximin's lofted pass before making an outstanding block. Limited the threat of Paraguayan winger Almiron, who was often forced to turn back to Manquillo for assistance.
4. Nathan Collins 7.5
Yet another solid showing from the 20-year-old Irishman, who has made himself at home in the Premier League. Stood firm in the face of Wilson's physicality and made a crucial intervention when flicking Almiron's shot over the crossbar.