Goals from Anthony Martial and a stoppage time winner from Marcus Rashford was enough to secure all three points for Mourinho's side.
The Spaniard, 29, replaced Fred in the 56th minute and gave his team greater stability from midfield, also going close to scoring as they pursued a winning goal.
Mourinho was impressed with his balance between defending and attacking, as Manchester United dominated during the second half.
"I was happy with bringing Ander to the game because normally the intention, when you look for a better result than you have, is to play attacking players or in theory to make attacking changes," he told the 'Daily Mail'.
"I felt that the team needed a player like Ander to give us balance in that position, to give us more pressure, recover the ball faster and also to protect Ashley Young, so I think I was happy with that change.
"The team was much better and we created lots of chances.
"As a team, the team performed much better in the second half. The two points we lost in the last minute against Chelsea, we got them back."
Callum Wilson had fired 'The Cherries' ahead in the first half, with Manchester United failing to impress.
"I think a lot of the aspects of our game were really good, especially during the opening period.
"We passed the ball well and there was really good intensity to our play.
"Manchester United changed things and made substitutions but I was pleased with my players' attitudes to the game and they gave everything to the match", he added.