It was the left-back's first Premier League start for more than seven months and was rewarded with a man-of-the-match award - just a week after picking the same award up against CSKA Moscow in the Champions League.
Romelu Lukaku's 25th-minute header was enough to seal the three points against Eddie Howe's men and Mourinho was pleased with how Shaw performed.
"Very good I think," Mourinho said. "Luke started so-so, which is normal but then improving, improving and improving, confidence levels going really high and he had a fantastic hour of football."
Does Shaw have a future in red?
The 22-year-old has been linked with a move away from Old Trafford with Tottenham Hotspur being one of the favourites to land him. However, after a number of impressive performances for United, his future could still well be in Manchester.
Shaw was replaced by Ashley Young and received a standing ovation from the home supporters, and Mourinho was thrilled at the way the supporters applauded his full-back.
"Physically he was feeling the cramps were coming, Eddie [Howe] was not far from me and understood that, so they tried to force a little bit from that side, I had to bring Ashley on to protect the team and Luke.
"The reaction from the crowd pleases me because there's not a better thing for a player than that," Mourinho added.
Young has been a regular starter at left-back in the last few weeks for United and Wednesday's clash against Bournemouth was the perfect opportunity to let Shaw prove himself.
With still five games to go in December, including a Carabao Cup quarter-final against Bristol City, Shaw will no doubt have another chance to feature for the Red Devils over the festive period.