United's leading marksman struck for the 11th time this season - but only the second time since December - as the Magpies ended a run of five league games without a win.
Alexander Isak had given Newcastle a 26th-minute lead with his fourth goal in six league starts for the club as the Swedish striker, who replaced Callum Wilson as Eddie Howe made five changes to the team beaten by champions Manchester City last weekend, deftly diverted home from Kieran Trippier's right-wing free kick.
Hwang Hee-Chan levelled in the 70th minute after a mix-up in the Magpies defence, but substitute Almirón won it nine minutes later as United leapfrogged Liverpool in the table and closed the gap on fourth-placed Tottenham, who have played two more games than Howe's side.
Isak had scored the only goal of the game when Newcastle last claimed three points with a 1-0 win over Fulham in January and came close to scoring in the 15th minute when Allan Saint-Maximin headed the ball on and Isak connected but couldn't direct his effort beyond visiting goalkeeper Jose Sá.
There was a let-off for United midway through the first half when Raul Jiménez pressed Nick Pope and won the ball from the Newcastle netminder before going down as he tried to take the ball past the keeper, with referee Andrew Madley and Video Assistant Referee Tony Harrington unmoved by the theatrical fall.
Those were the only incidents of note until Isak opened the scoring with a fine header from 12 yards, but Newcastle appeared determined to seize the initiative from there and almost had a second moments later. Saint-Maximin fed Isak and Wolves failed to clear his cross, with Joe Willock eventually seeing a close-range shot deflected behind by Craig Dawson.
Then, after Pope had saved a Rúben Neves strike and started a quick counter-attack, Isak had a goalbound effort blocked by Max Kilman.
Newcastle were pinning Wolves in their own half and in the 35th minute, a deep Trippier corner was hooked back across goal by Dan Burn and Bruno Guimarães headed against the crossbar.
The visitors sounded a warning shot four minutes later when Daniel Podence cut in from the left and his crisp low drive hit a post, but United continued to push for a second with Sá saving from Murphy at the end of the first half, then parrying a Willock shot from a narrow angle at the beginning of the second.
Half-time substitute Pedro Neto forced a good save from Pope with a cleverly-worked free kick before, after Nélson Semedo's cynical foul on Jacob Murphy, Trippier's dead ball strike was pushed away low to his left by Sá.
Pope had made a good save from João Moutinho midway through the half but was beaten in the 70th minute when Jiménez prodded the ball towards goal and Trippier slipped, allowing Hwang to sweep home moments after he was introduced from the bench.
But it was another substitute, Almirón, who had the final say. The excellent Willock opened up the visitors' defence and played in the Paraguayan, who curled home with a delicious left-footed finish from the right-hand side of the penalty area.
In stoppage time, after a great move, Almirón cut the ball back for Sven Botman to blaze off target, but it made no difference to the final outcome as United came out on top.