Deadline-day signings defender Christopher Samba, bought in a £12.5million deal from Anzhi Makhachkala, and midfielder Andros Townsend, on loan from Tottenham, both started, but it was Taarabt who provided the main threat for the hosts.
However, when his big moment came from the penalty spot just before the hour, the Moroccan saw Bunn produce a superb save to help earn Norwich what could prove a useful point in their own quest to avoid the drop.
Norwich - looking to build on their midweek draw with Tottenham - created an early opening when Robert Snodgrass got down the right and cut the ball back into the six-yard box where Wes Hoolahan hooked it over the bar, but the flag was up for what looked a narrow offside call.
With £8million striker Loic Remy out through injury, Taarabt proved QPR's most creative player and he drilled a low shot across goal and wide from the edge of the area on 10 minutes.
The Moroccan was again the instigator when his jinking run into the Norwich area put Jamie Mackie in on the angle, but his tight shot on the turn dragged the ball wide.
Norwich, meanwhile, were content to hit Grant Holt on the break, but the captain was often isolated and lacked support from midfield.
The visitors were forced into a change on 24 minutes when winger Anthony Pilkington hobbled off down the tunnel and was replaced by Elliott Bennett.
On 28 minutes Norwich got a fortunate break down the right as the ball ricocheted out to Hoolahan. The Irishman floated the ball over to Snodgrass, but his cushioned header was straight at Julio Cesar.
At the other end, Shaun Wright-Phillips turned marker Javier Garrido on the 18-yard box, but then skewed his shot wide.
Taarabt's inswinging corner dropped onto the top of the bar as QPR pressed at the end of the first half.
QPR again went close just after the restart when Taarabt's free-kick from the right flew across the face of goal when any sort of contact would have put the ball into the net.
Norwich almost snatched the lead on 53 minutes.
A long clearance was knocked on by Holt out to the right for Snodgrass and his curling cross was flicked goalwards by Hoolahan, but Cesar made a brilliant reaction save at point-blank range.
Samba then did enough to make Michael Turner sky the ball over the bar from the corner.
QPR made a change on 55 minutes when Bobby Zamora replaced Wright-Phillips.
The hosts were then awarded a penalty when Mackie latched onto a poor back pass from Garrido and was tripped by Bunn, who was shown a yellow card.
Taarabt went to place the ball in the bottom left corner, but the keeper had guessed correctly and pulled off a fine save at full stretch.
Bunn then beat away a dipping 20-yard effort from Taarabt as QPR remained on the offensive.
Cesar, though, was also in good form as he palmed out a 25-yard strike from Bradley Johnson as the match finally opened up.
Samba's glancing header, which looked to be going wide, was hacked off the line by Holt.
With 23 minutes left, Jermaine Jenas, signed from Tottenham on deadline day, replaced Mbia before a final change shortly afterwards as Tal Ben Haim came on for Fabio Da Silva.
Cesar pushed a long-range effort from Snodgrass behind before Bunn again impressed with a full-stretch save from Townsend.
Norwich sent on new man Luciano Becchio for the closing stages. The former Leeds striker looked to have been bundled over by Clint Hill in the six-yard box at a corner in stoppage time but referee Jon Moss thought otherwise.