The former England keeper somehow clawed away the Scotland international's 64th-minute header on a night when Redknapp saw plenty to persuade him that his cause may not be a forlorn one.
Green replaced the injured Julio Cesar, who had earlier made a vital block to deny Fletcher, at the break and stood firm as the visitors turned in a spirited display at the Stadium of Light.
Home keeper Simon Mignolet also had to be at his best to keep out Sunderland old boy Djibril Cisse as well as Jamie Mackie, and he was relieved to see midfielder Stephane Mbia blast a long-range effort inches wide.
Ultimately, Redknapp's men left the north-east still without a league win this season, but with a fifth point of the campaign which extended the Black Cats' difficult run to just two wins in 21 outings.
Given the circumstances, it was never likely to be a classic, and on a night when Wearside was lashed by wind and rain once again, there was little genuine quality on show for much of a tight contest.
However, there were chances and both sides could conceivably gone in at the break ahead.
Redknapp's men, of course, had an opportunity to try to persuade their new manager that they have a part to play in his brave new world at Loftus Road, and Adel Taarabt in particular seemed keen to take up the challenge.
He produced much of what was admirable about the visitors' attacking play to link well with frontman Cisse, and it was perhaps significant that for much of the half, it was Black Cats left-back Danny Rose who was Martin O'Neill's most enterprising performer.
Cisse might have got Rangers off to the perfect start when Taarabt picked him out unmarked at the far post with a fourth-minute cross, but his downward header lacked both power and direction and ended up safely in the arms of Mignolet.
Craig Gardner whistled a free-kick just wide seconds later, but Sunderland were forced into an early reshuffle after O'Neill's gamble on skipper Lee Cattermole's knee injury backfired when he limped off following a single, but robust 50-50 challenge with Samba Diakite.
Mignolet was extended much further with 12 minutes gone when, after Taarabt has twice combined with Esteban Granero, Cisse curled a right-foot shot towards the bottom corner and saw it tipped around the post at the last minute.
Cisse smashed a 19th-minute drive just wide, but as the game settled down, it slipped into something of a lull.
However, Belgium international Mignolet, whose howler against West Brom on Saturday proved so costly, had to make a vital block to keep out Mackie seven minutes from time, and he was indebted to his defenders as bodies were thrown into the path of follow-ups from Taarabt and Cisse as the Londoners sensed blood.
Fletcher got his first sniff of goal four minutes before the break after being played in by Adam Johnson, but he found Cesar equal to the task.
Rangers skipper Ryan Nelsen headed a later Taarabt corner straight at Mignolet, and it was the visitors who headed for the dressing room the more satisfied.
Cesar, who had picked up a knock during the first half, failed to re-appear after the break and was replaced by Green, and it was his side who threatened first after the restart when Granero headed Armand Traore's cross straight at the keeper.
The home fans were growing increasingly restless as QPR retained the ball with little difficulty, and their mood did not improve when Mbia was given time and space to drill a 25-yard effort just past the post with Mignolet at full stretch.
However, it was opposite number Green who had to demonstrate his enduring quality with 64 minutes gone when Fletcher met Sebastian Larsson's inswinging corner with a downward header which seemed destined for the net until the keeper somehow managed to block it one-handed and then clutch it to his body.
Redknapp replaced Diakite with Park Ji-sung seconds later, but it was Sunderland who threatened when Clint Hill was booked for a clumsy challenge on Jack Colback inches outside the penalty area, although Gardner's free-kick was blocked easily by Granero.
Larsson tested Green with a skidding strike from distance nine minutes from time, but the keeper got down well to claim it.
The locals begged their side to attack as the clock ran down with the visitors retreating, but it was Rangers who might have snatched victory at the death when Cisse, Granero and Mackie combined to set up substitute Shaun Wright-Phillips, only for Mignolet to block with his legs.