The Everton loanee saw his late penalty saved but reacted quickly to slam in the rebound and seal a crucial win for the Latics.
Baxter opened the scoring after 17 minutes, capitalising on a lapse in concentration from Pools goalkeeper Scott Flinders to poke home from close range.
The hosts were rewarded for upping the tempo after the break as Jonathan Franks' low drive on 65 minutes was spilled by Dean Bouzanis and Antony Sweeney tapped in.
Pools were reduced to 10 men on 76 minutes, Neil Austin red-carded for a lunge on James Wesolowski.
And the visitors, who hit the post on 88 minutes through Chris Sutherland, nabbed a late winner. Jack Baldwin handled in the area and, after Baxter's penalty was saved, he put the follow up away.