The Glasgow giants were eliminated by Czech minnows Jablonec in the Conference League play-offs on Thursday, meaning there will be no European group stage football at Ibrox for the first time since the 2017/18 season.
Lawrence Shankland's second-half penalty earned a much-needed win for McInnes to ease the questions over his future after just eight games in charge.
Rangers began the day already eight points adrift of rivals Celtic, but were rewarded for a more purposeful performance.
South Korean winger Kim Min-su was thrown straight in for his debut in one of four changes to the side beaten 1-0 on the night and then on penalties in the Czech Republic.
Min-su impressed, but it was Rangers' other winger Djeidi Gassama who won the decisive spot-kick 21 minutes from time.
Shankland sent Marius Muller the wrong way from the spot for only his second goal since joining from Hearts, both of which have come from the penalty spot.
Aberdeen skipper Kevin Nisbet thought he had equalised nine minutes from time but was judged to have pushed defender Dujon Sterling before heading home and the goal was ruled out.