Marcos Senesi had put the Cherries ahead at the Vitality Stadium, prodding home after a slow start from Eddie Howe's side on his first return to the club he left as manager for a second time in 2020.
But Almirón's coolly-taken leveller on the stroke of half time earned United a point on their travels, keeping them in fourth place in the table and just four points shy of second spot.
Though Fabian Schär nodded a Kieran Trippier corner wide of the target early on, the first meaningful effort came from the hosts. Jaidon Anthony's cross was met by Dominic Solanke and his glancing header looked destined for the bottom corner before Nick Pope threw an instinctive arm at it to turn it past the post.
Anthony swerved one well wide at the other end and just before the half hour, the hosts - who had prevented United from getting into the kind of rhythm supporters have become accustomed with this term - moved in front. Dango Outtara's flick-on from a corner found Senesi lurking at the far post, where he stabbed home from close range.
Howe responded by introducing Anthony Gordon in place of Joe Willock, who looked to be carrying an injury, before the break and though Alexander Isak - in for his first Premier League start in almost five months with Callum Wilson out - mustered the visitors' first shot on target just before half time, this was becoming something of a slog for Newcastle.
But just when it looked as though they were going to be going in behind at the interval, Howe's men struck against the run of play. Like so many of their best moments in recent weeks, it owed much to the industry of Sean Longstaff, who burst onto Allan Saint-Maximin's pass and was unlucky not to score himself. His well-struck low shot was repelled by Neto in the Cherries' goal but he could only push it away as far as Almirón, who rolled his strike back across goal and into the far corner to equalise.
Saint-Maximin blazed one over after the restart as the contest remained fairly open and end-to-end, though no real scoring opportunities were carved out until the final quarter of the game. One fell to Isak, who fired over the crossbar from an acute angle inside the box and may have been relieved to see the flag go up.
Another, moments later, fell Bournemouth's way, with substitute Marcus Tavernier shooting straight at Pope when well placed, before Saint-Maximin's cross-shot at the other end was spilled by Neto. Longstaff followed up but his effort was blocked, before Gordon saw his own attempt deflected past the upright.
But a point apiece was, in the end, a fair result. The Cherries may feel a little hard done by; they could have snatched it at the death when Trippier produced a crucial clearance off his own goalline to deny Solanke, who had backheeled Antoine Semenyo's low centre towards goal in what was probably the most clear-cut opening of the second 45 minutes, but it came and went as the game finished even.
