Virgil van Dijk gave Liverpool a first-half lead from a corner, before Dominik Szoboszlai eventually put the match to bed as he doubled the lead deep into stoppage-time to seal the three points.
Mac Allister started in midfield but went down in the opening minutes after a heavy challenge, playing with his left knee strapped until he was replaced just before the hour mark by Curtis Jones.
The 24-year-old has started 14 of Liverpool's 15 League matches so far this season, but he is now a doubt for Saturday's trip to face Crystal Palace in the lunchtime kick-off as Klopp revealed the cut was significantly hampering Mac Allister before he came off.
“Macca, it was a shame," Klopp said.
"Early, he stepped on his knee, there was a cut. As long as the wound was open, we just strapped it, obviously it was painful but he could play.
“Then we stitched it but then obviously it got a bit tighter there and then he couldn’t move anymore. Now we have to see how long that will take. I hope not too long.”
Liverpool will be hoping it does not prove to be another significant injury absence, having already lost Joel Matip for the rest of the season with a ruptured ACL. It adds to the problems facing Klopp, who is currently without the likes of Andy Robertson, Alisson Becker and Diogo Jota too.