Daizen Maeda gave Martin O'Neill's Celtic a first-minute lead at Hampden after an error from Saints stand-in goalkeeper Ryan Mullen, who subsequently went off injured, replaced by 17-year-old debutant Grant Tamosevicius.
Right-back Anthony Ralston added a second on the stroke of half-time, but St Mirren striker Mikael Mandron reduced the deficit and then took the game to extra time with a last-gasp leveller.
However, two goals from Kelechi Iheanacho, either side of a strike from fellow substitute Luke McCowan, and a finish from Benjamin Nygren in a ruthless six-minute spell in the first additional period, crushed the Paisley side.
O'Neill will face his former captain Neil Lennon in his final game in charge of Celtic next month after Dunfermline, who play in Scotland's second tier, beat Falkirk in a penalty shootout on Saturday after a 0-0 draw.
"If there is such a thing as enjoying it, I enjoyed the last three minutes of extra time," O'Neill told Premier Sports.
"Tough game, and it proved really difficult. Even at 2-0, it felt we wouldn't see it through. Towards the end, it felt they could get a second, as they did.
"But we found something in extra-time, and Iheanacho's goal turned it immediately for us again."