The former Everton and Oldham striker claimed his second goal in the space of a week midway through the second half as Nigel Clough's side stunned high-flying Swindon at Bramall Lane.
Town boss Mark Cooper was left infuriated by several of referee Scott Duncan's decisions throughout this hard-fought contest.
But, on the balance of play, could have few misgivings about the final result.
These two teams have experienced vastly different fortunes since reaching the League One play-offs last term.
But it was United, embroiled in a desperate battle against relegation at the wrong end of the table, who made the brighter start although their promotion-chasing opponents improved as the first half progressed.
Wes Foderingham was fortunate to escape during the early skirmishes after spilling a long-range attempt from Tony McMahon before Baxter's free-kick appeared destined to hit the back of the net until ricocheting to safety off the wall.
With Jamie Murphy, making his first start in two months, continuing to probe along the flanks, the visitors were firmly on the back foot until Nicky Ajose enabled them to establish a foothold in this absorbing contest.
The centre-forward, on loan from Peterborough, drew a smart save from George Long before creating another opportunity for Ryan Mason. Again, though, Long proved equal to the task.
United could count themselves unlucky not to take the lead after creating three fine chances in the space of sixty seconds almost immediately after the break.
First Baxter forced Foderingham to scramble clear with an angled attempt from the edge of the penalty area and, during the melee which followed, Ryan Flynn thought he had threaded a low drive through the crowd until the ball was diverted behind for a corner.
Foderingham was beaten from the resulting set-piece but Chris Porter's side-footed effort struck the foot of a post.
Nevertheless, United were not to be denied and got their reward just past the hour when Baxter pounced.
Aidy White delivered a deep cross into the Swindon box and when Flynn nodded it back into the danger zone, the Liverpudlian timed his jump perfectly to turn home.