The Buddies booked a place in the last four thanks to a 4-2 win on spot-kicks at Pittodrie after twice leading in normal time.
Sam Parkin and Kenny McLean's strikes were pegged back by Dons duo Scott Vernon and then, in injury time, Josh Magennis.
Lennon's men eventually eased through after Stephen Hughes hit the crossbar, and Cammy Smith saw his spot-kick saved by Craig Samson.
Lennon said "I'm a very proud man tonight. I thought the boys were like gladiators tonight. They fought right to the end.
"The character the boys showed throughout the game was first-class. To lose the lead in the dying minute of the game was very difficult to take and the boys showed great character to take us over the line and take it to penalties.
"If every game was like that, stadiums would be full across the country. I think both sets of fans were entertained throughout. It was a good game of football to watch."