Goals from Ryan Gauld and Ryan Dow ensured United of victory at Tannadice, despite substitute Liam Polworth having pulled a goal back nine minutes from time.
The result kept alive United's hopes of European football and manager Jackie McNamara said: "I think on balance we just shaded it. When you go 2-0 up you are comfortable.
"But they [Inverness] never really gave us anything to worry about until their lad unleashed that great strike towards the end.
"It's a great three points. We worked really hard for the win. We showed a lot of desire and guts.
"Ryan Dow chased a lost cause for his goal and got the rewards for that.
"We showed a good mentality here. I don't think there were many weaknesses.
"It keeps that winning mentality going, that's eight wins in nine games.
Looking ahead to the next fixture, Saturday's contest against Aberdeen, McNamara said. "It's another big game and hopefully we can get the three points.
"We have two lads suspended - Andrew Robertson and Nadir Ciftci - but we'll get on with it.
"If we win on Saturday it would give us a great lift - but there are still lots of games left between now and the end of the season."
Inverness manager John Hughes admitted his side had only themselves to blame for defeat.
He explained: "We've given the game away - that's for sure. That goes for the two goals.
"Our keeper (Dean Brill) apologised to the lads in the dressing room afterwards for the second goal. But that's the kind of guy he is.
"He trains like a world champion and has been outstanding since I came to the club. He's a fantastic character and has great spirit. He has kept us in plenty of games.
"I wouldn't like to think any of our players have downed tools for the season. We've still got plenty of cup finals to play."