Loanee goalkeeper Tom King made a number of good saves throughout for the hosts, most notably parrying Danny Grainger’s powerful free kick from 25 yards in first-half stoppage time.
United striker Sam Cosgrove shot straight at the Boro goalkeeper with an attempted chip when through on goal after 30 minutes, in what proved to be the game’s best chance.
King also kept out substitute Hallam Hope’s low effort within 60 seconds of the Carlisle striker coming on, then showed good reactions to claim Mike Jones’s deflected goal-bound shot late on.
Boro, who last lost at home on 21 October, threatened in the opening 10 minutes through Matt Godden and Tom Pett.
Godden flashed a low drive across the face of goal from a tight angle, while Pett’s glanced header from Ben Kennedy’s cross went wide.
Pedro stepped off the bench to send Chelsea into the FA Cup semi-finals after they beat Leicester 2-1 at the King Power Stadium on Sunday evening.
RB Leipzig came from behind at the Red Bull Arena to end a run of four winless games and defeat Bundesliga leaders Bayern Munich 2-1 on Sunday.