Huddersfield hit back midway through the second half with two goals in a three-minute spell.
Substitute Cristian Lopez headed the equaliser and Martin Patterson slid in at the far post to score the second.
Oldham levelled in the 71st minute when Baxter's free-kick was deflected into the path of Montano and the winger fired home from the edge of the penalty.
And Montano hit the winner five minutes later after Jonson Clarke-Harris' through ball sent him clear.
- Sports & Recreation
- Jose Baxter
- Adam Rooney