Jamie Mackie's first-half header helped the Hoops claw back a point after falling behind after just eight minutes to a long-range Brett Holman strike.
QPR pushed hard for a winner and came close on a number of occasions, yet they were unable to eke out a much-needed victory.
The 1-1 draw sees the Hoops equal the Premier League's longest ever winless start to a season, with their abysmal run only matched by the Swindon side of 1993/94.
However, Redknapp was encouraged by what he saw in his first home game in charge and believes his players can count themselves unlucky not to have won.
"We need a win, we need to get win on the board somehow," the QPR boss said.
"We had people out there like Clint Hill and Ryan Nelsen. Ryan came here and probably felt he was going to be fourth-choice centre-half and he is playing every week and is captain.
"They gave everything and Shaun Derry came on for us when I lost Stephane Mbia and did a great job.
"They tried for their life and I couldn't fault their effort.
"It was a bit like the game the other night when we drew at Sunderland.
"I was pleased with the performance and there was only one team that was going to win. I don't know really what they did after they scored.
"Villa started well and I thought we're in trouble here when we went one down, but we came back, got the equaliser and there was only one team creating chances after that and it was us.
"The woodwork saved them on a couple of occasions and they kept the ball in the end, passing the ball around in what were not dangerous areas.
"They were never a threat I didn't feel the longer the game went on."
While frustrated by the draw, there is good news on the injury front as Redknapp played down the issues that forced Stephane Mbia and Samba Diakite off.
"It's not too bad, he has just got a bit of a neck injury," Redknapp said of Mbia, who was stretchered off in the first half after a long break in play.
"It shouldn't be too bad. He has gone to hospital for a scan just to make sure he is okay, but they don't feel there is any problem there.
"Diakite has just got a little kick on the calf. Not too much wrong with him really."
Aston Villa counterpart Paul Lambert was the more satisfied manager after the match, having seen his wide secure a point from a potentially difficult trip to west London.
"I think it was a really tough game for us because of obviously Harry coming in," Lambert said.
"QPR, with where they are in the league, are fighting for every point and new manager syndrome normally kicks in.
"But I've got nothing but praise for the team, I thought they were excellent. They defended strongly when they had to and we were also a threat.
"The onus is always on the home team and for an hour or so I thought we were very good.
"I thought the first half especially we were excellent and I couldn't be more happy with the team and their progress."
While they defended resolutely, Villa had a lot to thank goalkeeper Brad Guzan for after the American produced a string of fine saves to deny QPR.
"He has brilliant the whole season, Brad," Lambert said.
"I am fortunate enough that football club has two fantastic goalkeepers in him and Shay Given."