Cole's first West Ham goal for 10 years earned the hosts a deserved point 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, and said: "It was difficult to turn Harry down - he's a top man and a great manager. I had another one (bottle) as well but I sent it to Harry for on the coach."
He added: "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."