Gallagher, who impressed in a loan spell at Crystal Palace last season, played 45 minutes on Saturday as Mason Mount sealed victory for Chelsea in the 83rd minute, after Timo Werner's opener was cancelled out by a Reece James own goal.
"Conor trains at a really high level, and you can clearly see it was the right decision to let him go on loan," Tuchel told reporters after the win. "He made a huge step in his development at Palace and became an England player.
"It was a very good start and a good match for him. Now it's important he gets used to our principles of the game, to understand when to move and when not to move, and his work rate is outstanding... He can be a big asset for us."
Chelsea will also take on Arsenal and Major League Soccer side Charlotte FC in the United States before finishing their pre-season preparations with a game against Serie A's Udinese in Italy.