Scott Robertson's first-half goal at Tannadice put Levein's men into the semi-finals where they will meet either Falkirk, Rangers or the winners of the Kilmarnock versus Celtic tie.
But it took a great stoppage-time save by United goalkeeper Lukasz Zaluska from a Stephen Glass volley to ensure a safe passage.
The manager said: "I was concerned that Saturday might play a part in this game, that was a huge factor.
"That's why I was relieved when we got through.
"Emotionally it was a very draining match for us at the weekend and I think Dunfermline coasted in the game because their game was called off.
"I was conscious of that but I thought defensively we were top-drawer.
"We managed to keep a clean sheet and that is five in the last six games.
"We didn't play our best but I don't think anyone could argue that we didn't deserve to win.
"I said to the players after the game that I didn't care about how we played, we're in the semi-finals and that's all that matters."



