The Buddies, beaten finalists in the competition in 2010, impressively ended a 14-game winless record against their opponents stretching back to August 2005.
Jamie Adams put through his own net at McDiarmid Park before Goodwin's excellent finish - his first for the Paisley men - gave Danny Lennon's side a deserved half-time advantage they never looked in danger of losing.
St Johnstone, who saw their five-game unbeaten run come to an end, were distinctly below par at McDiarmid Park given their impressive 3-0 victory at Motherwell at the weekend.
The hosts made one change from Saturday with Sean Higgins handed a first start in place of the injured Kevin Moon.
St Mirren, 2-0 losers at Hearts on Saturday, brought in Kenny McLean and Marc McAusland to their line-up. Derek McInnes' hosts were looking to continue their recent cup success, after five semi-final appearances in the last five seasons.
But instead it was the visitors who went on to dominate the first-half proceedings in some style. Stand-out midfielder McLean, 19, soon struck the palms of Peter Enckelman with a 20-yard drive.
Steven Anderson and Dave Mackay were then involved in a defensive tangle, with Gary Teale almost profiting as he chipped wide.
McInnes, serving the second game of a two-match touchline ban, saw his mood in the stand further darken when Frazer Wright hobbled off to be replaced by David McCracken.
The hosts finally threatened on 21 minutes when Francisco Sandaza wriggled free on goal after a Higgins pass, but his shot was blocked and Liam Craig's follow-up was weak.
St Mirren were soon back on the front foot, enjoying success on the flanks. Jeroen Tesselaar's cross picked out David van Zanten but his free header was held comfortably by Enckelman.
But the deserved breakthrough for the Buddies came on 39 minutes. Steven Thompson's header back across goal was set to be headed in by McLean, only for the luckless Adams to do the job for him. It was St Mirren's first goal in four games.
Then, on the stroke of half-time, Goodwin doubled the lead with a stunning finish, firing an angled 20-yarder that flew past Enckelman. McInnes immediately put on David Robertson and Willie Gibson upon the re-start to try and spark a comeback. Sandaza, Cillian Sheridan and Mackay soon combined, with the latter's effort held by Samson.
Gibson then picked out Sheridan from the left, but the Irishman's header was straight at the goalkeeper. St Johnstone continued to dominate second-half possession, but created little in the way of clear-cut chances. On the break, Teale saw an effort saved by Enckelman at his near post before Sheridan looked to have pulled one back on 81 minutes only for Samson to scramble his header away from danger following a Craig delivery.