United were largely outplayed by Celta Vigo in their semi-final second leg on Thursday at Old Trafford, but Mourinho's men managed to hang on for a 1-1 draw and triumphed 2-1 on aggregate.
And Keane, a former United captain who played for the club from 1993 to 2005, was not inspired by the performance.
"First of all, congratulations to United," Keane said on ITV, as quoted by Metro.
"But their level of performance tonight has just been letting a team like Celta Vigo dictate the game and keep themselves in the game in the last minute. That's not a good reflection on United.
"It goes to show where they've been all season, the amount of draws they've had. You can see tonight, just almost having that defensive mindset for the whole night."
Marouane Fellaini opened the scoring after 17 minutes Thursday before Facundo Roncaglia's 85th-minute strike pulled Celta within one goal of going through. They nearly pulled off the comeback, only to see John Guidetti miss a golden opportunity just before the final whistle.
Now United find themselves set to face Ajax in the Europa League final May 24 in Stockholm.
"A team like Manchester United at home to Celta Vigo? Go and get the job done, win 2-0 or 3-0 — it was there for the taking," Keane added.
"But they could not get to that level, which they've not done for most of the season. They need to be more ruthless. It’s not just tonight, it's been all season.
"For a team like Manchester United, with the players they have on the pitch, whether the mentality is coming from the manager, you can make lots of excuses.
"But they got away with it tonight, they've got to be better — be more ruthless."