Joe Shaughnessy was sent-off in the 13th minute for a reckless challenge on Zaine Francis-Angol, while the Dons suffered another setback when Clark Robertson was withdrawn with suspected knee cartilage damage.
But Motherwell could not make their numerical advantage count and substitute Andrew Considine headed Aberdeen ahead in the 83rd minute before Jonny Hayes wrapped up the win in injury-time.
McInnes said: "They have really answered up when the odds were stacked against them. We said to the players at half-time that they needed organisation, guts, character, everything you need to win the game, but the biggest thing was having a belief.
"To go so long with 10 men was always going to be a concern and with Clark going off as well, we only had one defender on the bench.
"It was gutsy, real guts and determination and a bunch of boys determined not to accept anything other than us getting into a semi-final."
McInnes accepted the referee's decision to send off Shaughnessy.
He said: "My only thoughts at the time and possible explanation for it was that Joe had slipped going into the challenge but I've watched it again and no complaints at all."
Motherwell manager Stuart McCall insisted John Sutton's early goal should not have been disallowed for offside, but gave Aberdeen credit for their stubborn defensive display.
He said: "I can't remember being this low and then to go into the dressing room and see the laptop and John Sutton's goal, which was onside, has just absolutely topped the night off.
"Full credit to Aberdeen, the way they defended with 10 men, they were outstanding.
"I was going to say the difference was we hit the bar from a set play and they scored from a set play but it's even more galling. Sutton is a foot onside.
"The linesman didn't put his flag up (initially), the delay was the fourth official telling him it hit (Keith) Lasley but it doesn't matter who it hits because it's onside."