Ronnie Moore's men snatched just their second win in seven outings at the expense of struggling Oldham to hoist themselves back into the top four having set a blistering pace earlier in the season.
The defeat saw the Latics' suffer their second successive loss since their fine FA Cup run came to an end at Everton, but Tranmere fans had little sympathy having witnessed their side recover from their own miserable run of results.
A fine one-handed save from Dean Bouzanis kept out Jean Louis Akpa Akpro's header as he stretched to reach Power's 13th-minute corner.
James Wesolowski's flailing strike failed to threaten for Oldham before Matt Smith headed straight at Owain Fon Williams.
It was Rovers who took the lead in the 61st minute, however, when Power delivered a strike that flew past Bouzanis from the edge of the box.
Tranmere had Fon Williams to thank when he scrambled Wesolowski's shot round a post.
Jose Baxter went closer to an equaliser when he rifled a 25-yard free-kick off the underside of the crossbar but Oldham could not snatch an equaliser as they failed to climb out of the bottom four.