United welcomed Liverpool to Old Trafford for the big game, and their rivals walked away with a comfortable 3-0 win.
Casemiro, one of United's older players, was responsible for two of Liverpool's goals, while Salah capped the game off in the second half.
Keane, speaking on Sky Sports after the game, didn't shy away from criticising United's performance on Sunday.
"You can try and dress it up a little bit, but United were shocking, really shocking," United legend Keane told Sky Sports. "Every time Liverpool went forward, they looked like they were going to score. You'd be worried for the manager. I would never stand here and say I expect a manager to lose his job, but the pressure must be building."
Casemiro came under major flak for his performance and was taken off at halftime by his manager Erik ten Hag. Keane took issue with the veteran Brazilian.
"Same old problems," the former United captain added. "Leopards and spots. Man United look very open; they got away with the first goal being disallowed. Mistakes from Casemiro, despite all his experience."
"They are all over the place. Casemiro must be aware of what's around him."
United now go into the international break and will return to action to play newly promoted side Southampton in 12 days.
Gary Neville, a former teammate of Keane, believes that Ten Hag will have to turn it around very quickly if he wants to keep his job at Old Trafford.
"Not to the point where his job would be under threat at this stage of the season, but there's no doubt he can't be mid-table or eighth, where he finished last season, going into Christmas. He just can't be.
"There has to be improvements. I think they'll judge him in a few weeks when all the players are fit, when Højlund's back. But there is some pressure, and today was a big game for United. It's a big game for Liverpool as well, obviously, but for Erik ten Hag, I don't think it's a game he can afford to lose."
United play Southampton on September 14, Saturday. That game kicks off at 12:30
