New signing Kagawa, who was named as man of the match, found the net after 68 minutes against a second-string Shenhua outfit, who lined up without former Chelsea players Nicolas Anelka and Didier Drogba at the Shanghai Stadium.
United had the first chance of the game after 25 minutes when Anderson chipped a through ball to another new arrival Nick Powell, who brought the pass down inside the box with his right foot before scuffing his shot wide with his left.
It was the hosts who drew the first save from either goalkeeper as Joel Griffiths broke through the middle before forcing a good stop from Anders Lindegaard who deflected the Australian's effort wide.
Kagawa - playing just behind a lone striker- was a thorn in the side of the hosts and used neat footwork to break free of the Shenhua midfield before sliding in the largely ineffectual Federico Macheda, who could not beat Shengjiong Qiu from inside the area, the home goalkeeper pushing the Italian's effort wide.
The visitors started to dominate and Bebe - who replaced Powell at the break- fired hard and low to force a good save from Qiu.
United did however take the lead when Bebe beat his man with ease on the left before playing in Japan international Kagawa whose effort deflected past Qiu for what would prove to be the match-winner.