Redknapp returned to the Dean Court dugout for the first time since 1992 and despite his side enjoying the opening stages of the Sky Bet Championship game, QPR returned to west London beaten after strikes from Lewis Grabban and Tommy Elphick.
Armand Traore scored a brilliant leveller for the away side but despite the Cherries having Harry Arter sent off with 20 minutes remaining Grabban's winner ensured Leicester will be playing top-flight football next season.
Richard Dunne came close to a late equaliser but QPR boss Redknapp said his side need Charlie Austin, who came off the bench for the first time since January, back if they are to challenge in the play-offs.
Redknapp said: "It is almost impossible now to be fair. Nobody is going to catch Burnley now are they?
"I thought we started great and played some really good football and then we began to give it away. I thought we would go on to win.
"We have to be careful with Charlie and we will see whether we will chance him again on Tuesday. He is the one guy who looks like he will score a goal. We are short of goals - that is our problem.
"We lacked that bit of flair to open them up, you are looking at a bit more from Junior Hoilett and Ravel Morrison, your flair players to open them up - and they didn't really do it."
The Hoops started the game well and former England midfielder Jermaine Jenas forced Lee Camp into a smart save on 15 minutes.
Elphick nodded in an Ian Harte corner just before the break to give Eddie Howe's men the lead but Traore expertly took a Tom Carroll ball in his stride to fire under Camp seconds after the interval.
Grabban struck his 20th of the season on 59 minutes to hand the Cherries their seventh win in nine matches and leave them five points adrift of a top-six place.
A delighted Howe said: "He has had an excellent season for us and hopefully there is more to come.
"It is a terrific achievement at this level when you look at the names he is up there with and I still feel there is more to come from him.
"His work-rate again today was a real feature for me and he was still chasing their centre-halves and he deserves this special moment.
"I cannot deny there is still a chance for us to get in the top six and Tuesday against Reading is going to be a very special game.
"I don't think it was a red card. It is not a tackle Harry had to make but it was not two-footed."