Earlier, a slip by Georges Santos in the third minute gave Jerome his first run on goal but his shot was smothered by Generoso Rossi.
Neal Ardley's accurate set-pieces were a constant danger and he set up James Collins, but the Bluebirds defender's header went high over the bar.
An excellent run down the right flank by Jerome led to City's goal in the 27th minute. He dummied past Santos, skipped past Danny Shittu and crossed into the penalty area where McAnuff was waiting to rifle a low shot across Rossi and into the far corner of the net.
Four minutes later McAnuff almost doubled the lead when a first-time shot whistled just past the post as the Bluebirds kept up the pressure.
It was five minutes before the interval until Rangers first troubled the home defence. Martin Rowlands made space before curling in a shot that Neil Alexander clutched at full length.
Rangers should have equalised early in the second half when a mistake by Darren Williams left Kevin Gallen in the clear but the Rangers skipper pulled his shot wide with only Alexander to beat.
Ian Holloway's side were far more determined in the second half and with City guilty of giving the ball away too easily it meant for long period that the Bluebirds were deep in their own half, desperately defending their slender advantage.
Midway though the half Peter Thorne and Jerome were replaced by Alan Lee and Paul Parry and ten minutes from time Gino Padula's free-kick rattled the Bluebirds crossbar.
Tony Vidmar came on for his last game in City colours and was on hand to head clear a Padula cross as Rangers pressed for an equaliser.
To their credit the visitors kept pouring forward, looking to end the season with at least a point but the Bluebirds defence held firm.