The Pittodrie team fell behind to strikes from Miles Storey and Ryan Christie but pulled one back before half-time through Ash Taylor.
They gave themselves too much to do, however, and despite chances for Peter Pawlett and David Goodwillie towards the end they were unable to level.
After an eight-match winning run which has taken them to the top of the league, Aberdeen have now lost twice in four days.
They were also beaten on Wednesday when they surprisingly suffered a 2-0 reverse against Hibernian in the League cup at Easter Road.
As was the case then, McInnes had no complaints about the final scoreline in the Highland capital after seeing his men struggle to contain the hosts.
Aberdeen do still have a four-point advantage over second-top Celtic after they could only draw 0-0 at home to Hearts.
McInnes said: "In the first 45 minutes I didn't recognise my team. We were second best all over the pitch.
"In my time here, with the disappointment of semi-finals, it's the poorest first half I've seen. Ryan Christie lit the place up.
"We never got to grips with him. Even though we dominated the second half, I feel the better team won the game."
Source: Sky Sports