Cole's first West Ham goal for 10 years earned the hosts a deserved point on Saturday as Loic Remy was denied a dream debut for QPR, who were cast four points adrift at the foot of the standings.
Cole was tracked by Redknapp - who signed him for West Ham as a schoolboy - but he opted for a second spell at the Hammers earlier this month as his disappointing spell at Liverpool came to an end.
"It was difficult to turn Harry down - he's a top man and a great manager," said Cole, clutching the man of the match champagne.
"I had another one (bottle) as well but I sent it to Harry for on the coach.
"I asked somebody to send it over on to the coach for him. I'll always be indebted to him for everything he's done and it's just a little gesture so he can have a drink on the way back to Loftus Road."
Redknapp, whose side are unbeaten in 2013, a run of five games, said: "I love Joe. I signed him when he was a little boy.
"I just wished him all the best. Of course I did. I hope he does great."
Redknapp is also demanding improvement from his own side as he prepares to pair Remy up front with Bobby Zamora, who is close to a return to fitness, with playmaker Adel Taarabt in behind. Remy's arrival has seen Djibril Cisse join Qatari club Al Gharafa on loan for the rest of the season.
The QPR boss added: "It will give us a bit of attacking threat. Taarabt could play in a diamond, in behind.
"I'll find a way of trying to get the best out of them."
West Ham are also optimistic of a greater cutting edge after struggling to breach QPR's defences despite being on top for much of the game.
Cole said: "We're just looking at each other in the dressing room - 46 crosses into the box, we like to get in the box with Marouane (Chamakh), Carlton (Cole) - and Andy (Carroll) to come back as well.
"If you get 46 crosses in and get those boys playing you'll have an opportunity to win games."
Chamakh had two good headed opportunities saved by Julio Cesar before the Brazilian goalkeeper could only parry an effort from substitute Carlton Cole, which Joe Cole tapped in.
"It's always nice to score a goal," said former Liverpool playmaker Cole, whose most recent strike for West Ham came on January 11, 2003, in a 2-2 draw with Newcastle.
"I'm enjoying my football, I feel I'm getting stronger. You can do as much training in the world but you need games.
"I feel I'm getting a bit stronger every game and it's nice to be affecting the game."
There was booing when Chamakh was replaced by Carlton Cole.
West Ham boss Sam Allardyce said: "They (the fans) were having a boo at me before half-time. It was my decision to bring him off (that resulted in the booing), there was no doubt about that.
"The West Ham fans get as frustrated as we do, but they can't knock the performance."
Chamakh will be ineligible to play his parent club Arsenal on Wednesday, with midfielder Cole eager to improve the Hammers' current run of one win and five points in seven Premier League games.
"You can have difficult moments in the season and if you can pick up points in the difficult moments you do all right," he said.
"I think 38 points will be enough this year and we want to get there as soon as possible. Let's focus on the next game at Arsenal - we want to take something from that - then you go from there.
"We've just got to go there and stifle Arsenal a little bit and then maybe we can do something."
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