Albert Adomah had put the Robins ahead only for Ishmael Miller to stab home from a corner.
Then at 1-1 Grant Leadbitter's corner was spilt by City goalkeeper Tom Heaton and seemed to go over the line before being hooked clear - and the decision was 'no goal'.
And City scored two more on the counter thanks to late strikes from Stephen Pearson and substitute Steven Davies.
McInnes said: "We played well at times in the early part of the campaign but that was as good a performance as we have put in all season. I was delighted for the lads and for the supporters.
"We expected Middlesbrough to attack us with the two men up front and we knew it would be tough, we knew we would have to work hard and keep our shape.
"We did that. I thought we contained them well and I thought we controlled the game for long spells and with that came a bit of confidence.
"That showed in how we attacked. We scored three very good goals. We got into some very good areas and created some good chances."