Goals from Rob Jones and David Cotterill in the space of four first-half minutes set Doncaster on their way to a 3-1 win and although teenager Dave Symington pulled one back for Carlisle with a superb free-kick after the break, Tommy Spurr pounced for visitors' third to condemn Abbott's side to a sixth home defeat.
Abbott, whose side lost 4-1 at home to Leyton Orient on Saturday and have now conceded 25 goals at Brunton Park this season, said: "I'm a manager under extreme pressure and I'm not going to hide from it.
"I can't drag any more out of the players than we did here. The fans were terrific and I can't ask for any more from the players.
"I thought the goals were extremely poor, but we got lots of balls into the box and the kids were up for the battle and gave everything they had.
"We had three 18-year-olds on the park up against vastly-experienced opponents, but at 2-1 I felt we still had a fighting chance and we aren't getting the breaks at the moment."