City struggled to find any form of fluency for much of the 90 minutes, and had to survive numerous chances squandered by their south London guests.
In the end, Maynard popped up in the third minute of second-half stoppage time to slide the ball beyond Steve Mildenhall in the Millwall goal, and McInnes admitted: "We said prior to the game that we'd sacrifice a performance for a result and I think that was possibly the case."
He added: "I think we've been better in some games over the last few weeks and not got what we thought we'd deserved.
"Other than the opening 20 minutes where I thought we came out right out of the traps and should have got ourselves in front, for me Millwall - being the away team - got far too much of the ball and enjoyed the game too much.
"In the second half there was a real determination at least not to lose the game. We were a bit more solid and I never thought we were going to lose the game.
"We huffed and puffed, and we were nowhere near at our best, but thankfully we kept going to the death and got our goal.
"It's harsh on Millwall, I've got to say that, but we've been the victim of that as well when we've not got quite what we've deserved. I think that was the case tonight. The result was the most important thing."
Millwall boss Kenny Jackett rued his players' lack of accuracy in front of goal, saying the Lions' struggles up front are becoming a worry, saying: "We had a lot of chances but that's been us, we haven't scored enough goals generally," he said.
"When the goal came at the end there it's cost us and it's cost us badly."