Spurs failed to put pressure on their rivals for a Champions League place following a Premier League stalemate at the Stadium of Light.
Their goalless draw was enough to nudge Spurs at least temporarily up to third, but it was a frustrating afternoon for Redknapp with Emmanuel Adebayor and Rafael Van der Vaart also missing chances.
Sunderland threatened in patches with a goal-bound Stephane Sessegnon effort striking the legs of William Gallas, and a brilliant Kyle Walker interception denying a simple chance for James McClean.
Redknapp's hunger for a win was obvious in his late introductions of Aaron Lennon and Jermain Defoe.
Adebayor, Bale and Van der Vaart all came close for the visitors who started well but faded as Martin O'Neill's resolute Sunderland held firm.
In a game of few clear-cut chances, the best for Sunderland fell to Sessegnon and Nicklas Bendtner, but there was precious little to threaten either of the two goalkeepers.
Given the attacking riches on show that came as some surprise, and goals certainly seemed to be in the offing in the early stages which the visitors largely dominated.
Benoit Assou-Ekotto saw his fierce low drive deflected wide, then Adebayor nudged a Van der Vaart cross goalwards but it was deflected wide by Phil Bardsley.
Sunderland had little to show for their efforts until a 33rd-minute mistake by Sandro threatened to gift Sessegnon a run on goal, but the midfielder responded with a fine saving challenge.
As Spurs' early pressure gradually subsided, the hosts had a loud penalty appeal for a handball against Scott Parker turned down by referee Chris Foy.
Bendtner almost turned in a Bardsley drive but was well smothered by Brad Friedel, and moments later Craig Gardner's header from a Bardsley cross was easily held by the keeper.
William Gallas stumbled attempting to clear a James McClean cross in the 39th minute and with the Black Cats continuing to press, McClean headed over from a Sebastian Larsson cross.
Van der Vaart struck a goal-bound effort in the last minute which was blocked by Matthew Kilgannon as the two sides ended a sporadically entertaining first half on level terms.
O'Neill's men almost made a dramatic start to the second half, McClean finding Sessegnon in the box and the striker's low shot striking the legs of Gallas.
The impressive Jack Colback, playing in an unfamiliar full-back role, denied Adebayor a close-range chance in the 52nd minute when he blocked a ball into the box by Modric.
Adebayor was caught narrowly offside as he attempted to scissor-kick home a lofted ball from Van der Vaart, then Bale headed onto the roof of the net from a cross by Assou-Ekotto.
Sunderland fashioned a fine chance on the hour when Larsson's low cross from the right looked set to give McClean the simplest of tap-ins before Walker made a crucial interception.
Van der Vaart cut a furious figure in the Sunderland box in the 73rd minute when his shot was blocked by Michael Turner, with the Dutchman almost alone in claiming a handball.
Walker's foul on McClean gave Sunderland a dangerous free-kick on the left in the 85th minute, but Larsson's set-piece was powerfully headed away by Younes Kaboul.
Bale's last-minute charge goalwards was blocked abruptly by a superb challenge by Gardner, following which the Welshman hobbled gingerly to the sidelines.