United pressed high early on, forcing Keating into the mistake for the opener, and while Liverpool recovered well the damage had been done.
Keating made some amends with two important saves to keep her side in the game.
Liverpool have conceded four first-half goals in five games, with only Charlton (five) conceding more in the league this season.
The Reds had an expected goals total of 1.41, compared to 0.59 for United, but were unable to make the most of their chances.
Olsson, Kapocs and Marie Hobinger all had two shots on target but could not get the better of Tullis-Joyce as they fell to their second league defeat of the season.
Liverpool boss Gareth Taylor said: "Their keeper was busier than ours. We make an error early and their tails are up. My early analysis? It's difficult when straight after but we need to be braver.
"We ended up becoming what they are, very direct. Their goalkeeper was literally kicking out of her hands. It's a very direct way of playing. We fell into that trap ourselves.
"We have got to be braver, to handle the ball and do what we do. It feels like the game last season where we tried to play a team at a different style. It surprised me today."