Rangnick has hailed his players for their "physicality", with the match coming just three days after a clash at Brentford, and the German was also keen to pay tribute to the supporters for their role in the result.
"The atmosphere (in the dressing room) is amazing. Quite rightly the boys were celebrating. They know what a massive win this was. Those are the best kind of wins when the other team has not time to come back. We are extremely happy with the performance and the level of physicality we played with bearing in mind we played three days ago at Brentford," Rangnick told BBC Sport.
"We didn't always find the best solutions in the final third. In the last 20 minutes we had too many shots on goal when there was no realistic chance to score. We needed to be a bit more patient, but in the end it was about taking all three points.
"We had to take some risks in the last 15 minutes but in the end I wanted to show the players it is about winning this game and I'm more than happy we scored the goal in the last minute. The red army's support during the game was amazing. I can only say thank you to all the fans.
"It's always good but today was massive and they were also aware of how important this game was. They pushed us through the whole game."
Man United are not back in action until February 4, when Championship side Middlesbrough visit Old Trafford in the fourth round of the FA Cup. body check tags ::