A brace from Marcus Rashford in the first half put Manchester United in control but an own goal by Eric Bailly in the second half threatened to make the game interesting.
Liverpool had a late appeal for a penalty waved away and Mohamed Salah shot over when in space.
But the win put United five points clear of their north-west rivals and Mourinho insisted his side deserved the victory.
Mourinho said: "We can split the halves - United first half and Liverpool second but in my opinion our first half was a half with goals and danger.
EPA
Marcus Rashford scored twice for Man Utd
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"In the second half - in my opinion, and probably the people in the studio have a different opinion - was Liverpool controlling with the ball and United controlling without the ball.
"Our team was always in control. Even in set pieces, corners, dangerous situations we were in control so I think we deserved it.
"Against Liverpool if you play bad when you have the ball you can be in trouble. In the second half it was not our intention - I cannot say this was the plan. Liverpool pushed us into defensive situation but we kept control. It was a complete performance by us with two different halves.
"If people don't think we deserved it, I don't care. I am a bit tired, we have a match on Tuesday. I don't care what people say. The boys are happy, I'm happy."