Scott McTominay's eighth minute winner was enough to confirm United's place in the next round of the competition. Waiting for them in the fourth round are Middlesbrough who they will face in early February.
The Red Devils kept their first clean sheet in four games as Aston Villa saw two goals ruled out in the second-half.
The second of those decisions was indisputable as Danny Ings was caught in a clear offside position before going one-on-one with David de Gea.
However, the first was far more controversial as referee Michael Oliver consulted the VAR screen before ruling that Villa's Jacob Ramsey, who was in an offside position, had obstructed Edinson Cavani in the build-up to Ings' goal.
Despite the controversy, Rangnick was pleased to see his side return to winning ways.
"We knew it would be very difficult against Villa. In the first half I thought we did well. It could have been 3-2 or 4-3 to us at half time.
"In the second half we were on the counter attack and we could have made more of those situations. We took the wrong decisions at times but when we changed to a diamond in midfield, we had more control.
"We fought well as a team together when it was important. In the end it was a good win. That was our goal, to proceed to the next round."
Rangnick said: "We have to develop the team into a clean sheet team and that was another one today but still David de Gea was outstanding again for me today.
"I would have wanted him to have less saves in total, I think we had four or five shots on goal but again we can still improve even in that area."
United will be looking to replicate the result on Saturday when they travel away to Villa Park for a Premier League clash with Steven Gerarrd's men.