St Mirren had lost 1-0 to Livingston at Hampden but ran riot at Tannadice.
The Buddies established a three-goal lead in the first half thanks to a Joe Shaughnessy header and two Jamie McGrath penalties.
United gave themselves some hope with Ian Harkes' long-range effort but the Paisley men scored another two late on through substitutes Dylan Connolly and Kristian Dennis.
Goodwin said: "It would have been very easy for the players to feel sorry for themselves after Sunday. But credit to them for a great reaction.
"I think the scoreline flattered us a bit but the first goal was massive – and it was a great set-play.
"I don't think we had anything to prove after Sunday. It was a sickener, there's no getting away from it. But we refocused and got right back at it in the league.
"I'm just really pleased for the players and staff because Sunday was so hard to accept. We've hopefully given the fans a few goals to cheer at home, as well."
United boss Micky Mellon admitted his side had given themselves a mountain to climb and were punished for some naivety.
He said: "We are disappointed to lose five goals when we have been defensively very solid.
"Going in 3-0 down at half-time, you give yourself a mountain to climb and we just said to them just keep playing and try to score the next goal and see where it takes you.
"We did with a cracking goal but we gave away another two cheap goals.
"I keep saying how inexperienced we are at times at this level. Tonight we showed naivety that we will learn from."