The north London club failed to qualify for Europe this season, but Redknapp is hoping to achieve success in one of the cup competitions and saw his Spurs side record a comfortable victory at home, despite missing several key players through injury.
Everton boss David Moyes was left fuming over the timing of the game following his side's Europa League exploits and he will be even more angry after seeing his side crash out of the competition.
The visitors were missing ten first-team players through injury, but almost took the lead after just four minutes when Louis Saha, with his back to goal, tried an audacious lob over Heurelho Gomes and the Spurs keeper responded with a superb save to tip the ball over the bar.
Roman Pavlyuchenko was making a rare start in the Tottenham attack and came close to getting his name on the scoresheet on 24 minutes.
David Bentley dummied the ball into the path of Alan Hutton, who raced away down the right-hand side and crossed in for the Russia international, who smashed the ball just wide.
The home side continued to create the best openings and had a golden chance to break the deadlock on 29 minutes. Benoit Assou-Ekotto played a high ball over the top of the Everton defence and Sylvain Distin made a total mess of clearing the ball
Keane reacted the quickest and lifted the ball over Tim Howard, but his effort came back off the post.
Tottenham got their just rewards when they opened the scoring on 31 minutes.
Bentley's cross into the box fell to Huddlestone, who made no mistake firing a right-foot half-volley past the stranded Everton keeper.
Everton boss Moyes was forced to make a change at half-time when he brought on Jo to replace the injured Saha.
The Brazilian almost made an immediate impact just 60 seconds into the second half when he laid the ball in to Tim Cahill, whose powerful shot forced a fine save from Gomes diving away to his right.
Jo was involved in the heart of the action once again on 55 minutes when he broke clear of Sebastien Bassong and was clean through on goal, but his low shot was superbly saved by Gomes.
Everton were made to pay as Keane was brought down inside the box by Distin just moments later.
The Tottenham skipper picked himself to take the penalty, but saw his spot-kick saved by Howard.
Keane and Gareth Bale both saw their shots blocked, but the rebound eventually fell to the Republic of Ireland striker, who made no mistake in striking the ball into the roof of the net.