The Premier League champions recovered from three consecutive defeats - including an FA Cup loss at home to the Welsh club - with goals from Antonio Valencia and Welbeck giving them a vital victory.
Three points were not enough to move United up the table, but they maintained ground on the likes of Tottenham and Everton in the fight for fourth.
Kagawa started the game on the left wing before swapping with Adnan Januzaj, who had been featuring in the hole behind Welbeck, at halftime. David Moyes' tactical tweak made the difference with the Japan international orchestrating United's play, and Welbeck hailed the former Borussia Dortmund playmaker's showing.
"I thought Shinji was the difference in the second half," he told MUTV. "Once we moved him to the middle he looked like he was at home and he really switched it on for us. He made a massive difference."
Welbeck's goal was his sixth in as many Premier League games, and the striker is hoping his colleagues can find a similar level of consistency to propel them back into the league's upper reaches.
"We've set a platform now and really want to build on that and gather some momentum," he added. "The next match is the most important now. That's what you've got to do, take it one game at a time."