The Togolese striker, who is currently at Spurs on loan from Manchester City, set up Rafael van der Vaart for Spurs' opener before Gareth Bale doubled his team's lead shortly before the half hour mark.
An early second half goal from Mohamed Diame pulled the hosts back into the game but Redknapp's men held on to make it three consecutive wins in the Premier League.
Adebayor failed to find the back of the net for the first time since joining Spurs towards the end of the summer transfer window, but Redknapp has stated he would want to sign the forward on a full-time basis, though his wages may prove to be a stumbling block.
When asked if he would like to sign the forward permanently at the end of the season, Redknapp responded: "Yes, if he keeps going like he's going. I'd love to keep him around, he's been excellent so far so let's hope he keeps it up.
"I don't know [if it will be possible]. We don't pay his wages and I think that would be the problem. God knows what he earns, I don't even know what we pay towards his wages, only the chairman would know that. I bet he's not cheap."
The victory at the DW Stadium was Tottenham's third in a row in the league after picking up three points against Wolves and Liverpool in recent weeks but despite being pleased with the result, Redknapp admitted his side had been sloppy to allow the home side back into the game shortly after half-time.
He said: “It was complete domination in first half, we could’ve been three or four up and I said ‘the only way you’re going to give them a way back into the game is if you do something stupid and give them a goal because it will change everything.’
"We gave a sloppy goal away, tried to take a liberty playing it out of the box and got punished. It changed everything, it gave them a lift, the crowd got behind them from giving them stick and they ran their socks off.
"In all honesty, our keeper’s not had too much to do but it still got to close for comfort in the end."
Spurs face north London rivals Arsenal next weekend with Redknapp pleased with his side's recent form and expecting them to improve in the coming months after overcoming a poor start to the season that saw two defeats in their first two games.
"The ship was never unsteady," he said. "We’ve got a good squad, a strong side and we’ll only get better.
"We’ve got some key players to come back into the squad as well. It’s three good points again, nine out of nine now."