Danny Swanson opened the scoring from the resulting indirect free-kick and Craig Thomson sealed the win late on.
Dons boss McInnes said: “It’s a poor decision from the referee to declare that a pass-back and they’ve taken full advantage.
“We probably didn’t do enough to get ourselves in front, but we certainly didn’t look like losing it and I think the referee has a part to play in the first goal.
“Our set-up in how we dealt with the free-kick is up for question, but I don’t think it was a blatant pass back.
“Once we go behind we throw bodies forward and we get done on a counter-attack, which can happen.”
St Johnstone are just the third team to leave Pittodrie with three points in this season’s Premiership alongside champions Celtic and third-placed Rangers.
Perth gaffer Tommy Wright said: “It was windy and blustery and the pitch is a bit uneven, so it wasn’t a game of beauty.
“We’ll play a lot better with the ball, but one thing we do when we’re not playing particularly well is grind it out.
“It’s up to the opposition to break you down and I don’t think Alan Mannus has had a save to make all game.”