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for a reckless challenge on Zaine Francis-Angol, while the Dons suffered another setback when Clark Robertson was withdrawn with suspected knee cartilage damage.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."
Source: Aberdeen Mad