QPR remains without a victory after 14 games but is now four points adrift at the foot of the standings after second-bottom Reading lost 1-0 at Aston Villa in another match between two of the division's strugglers.
Belgium striker Christian Benteke headed in Villa's winner in the 80th minute, earning his team its third win of the campaign under new manager Paul Lambert. Villa climbed out of the bottom three and is now level on points with Sunderland.
The poor fare on show Tuesday suggests all four teams will be fighting against relegation this season, although QPR will hope the appointment of Redknapp will spark an upturn of fortunes in the coming weeks.
"There were some good points coming out of that. I felt more optimistic tonight than I did on Saturday," said Redknapp, who took the reins after a 3-1 defeat to Manchester United at the weekend. "I just have to keep encouraging them."
Redknapp has previously worked wonders at Tottenham and Portsmouth when taking over at a struggling club, but he acknowledged Monday that the salvage operation at Loftus Road is his biggest challenge yet.
On this evidence, Redknapp's team will be harder to beat than that of previous incumbent Mark Hughes, who lost his job on Friday with QPR the only side in all of England's four leagues yet to have won a game.
Redknapp fielded a more defensively minded midfield at Stadium of Light and saw his team create only a few chances, with Jamie Mackie wasting the best of them in a one-on-one opportunity in the first half.
Sunderland was just as rudderless at the other end, however, and was booed off by fans who are growing increasingly impatient with manager Martin O'Neill, who was forced to deny rumours circulating this weekend that he had offered his resignation.
Steven Fletcher had the hosts' best chances but was first denied by Julio Cesar and then Rob Green, who replaced the Brazilian in goal at halftime.
With Darren Bent left out of Villa's squad for the second match running, the onus was on Benteke to carry the goal threat and the 21-year-old striker delivered the goods by planting a downward header into the net from Ashley Westwood's left-wing corner.
"It's massive," Villa midfielder Barry Bannan said. "We need to win games and it's a big relief for us. There's been a lot of people talking about us being in the bottom three and it's hard to take."
The focus turns to the top of the table on Wednesday, with the two Manchester clubs and Chelsea in action in the remaining eight midweek fixtures.