In a game which only sporadically sprung into life, QPR goalkeeper Julio Cesar made smart stops in the early stages of each half to ensure Redknapp's new club moved off the foot of the table for a few hours until Reading's victory over West Brom.
Tottenham could have been in front after just five minutes, but were denied by a tremendous double save from Cesar.
The Brazilian firstly tipped Jermain Defoe's thumping drive onto the post before recovering quickly to get down low to keep out Emmanuel Adebayor's shot from the rebound when the Togo striker should have scored.
QPR responded through Shaun Wright-Phillips, who saw his shot deflected past the upright by Michael Dawson after the winger had been played in by a superb pass from Adel Taarabt.
But chances were scarce for the rest of the opening 45 minutes as Spurs were forced to bring on Scott Parker for the injured Sandro before the half closed with Wright-Phillips shooting wide after he had skipped past Kyle Walker and Dawson.
Spurs started the second half strongly as Gareth Bale's swerving free-kick was parried by Cesar, who could then only watch as Walker's arcing free-kick dipped just over the crossbar.
The visitors continued to carry the main attacking threat and Defoe found a yard of space in the box on 53 minutes only for the England international's shot to clatter off Cesar.
Hugo Lloris had been a virtual spectator in the second half, but the Spurs keeper was out smartly with 13 minutes to go to clear the danger after Taarabt had again tried to slip Wright-Phillips in behind the defence.
Taarabt and Wright-Phillips linked up to good effect again with five minutes to go, only for the former Chelsea winger to stumble in the penalty area as he prepared to shoot.
Spurs tried to exert pressure in the final few minutes, but their passing lacked incision as QPR comfortably held out for a deserved point.