Dos Santos scored just his third goal in almost four years at the north London club to help Redknapp's team ease in to the fourth round of the FA Cup.
The Mexican, who has had spells at Racing Santander, Ipswich and Galatasaray during his time at Spurs, looked set for a disappointing afternoon when he saw Jermain Defoe effectively steal a goal off him when the England striker coasted in to the box and tapped home the youngster's goal-bound effort from barely a yard.
The 22-year-old bounced back in style, however, helping Defoe set up Roman Pavlyuchenko for Tottenham's second before adding a third of his own at the death with a clever lob.
Dos Santos infuriated Tottenham by turning down a £13million move to Udinese last summer as he wanted to go to Sevilla, and Redknapp still thinks the forward lacks the determination to make it as a top-flight player in England.
For that reason, the Tottenham boss expects Dos Santos to leave White Hart Lane despite his exploits.
"I think Gio will move if we get the right player (to replace him)," Redknapp said.
"He's looking to move to be honest. We had a big bid from Italy last summer but he wants to go back to Spain.
"He needs to play and to be fit and train hard, and when he does that, he's got a chance.
"He's got talent, he's got ability, he just needs to work harder in training and get fitter.
"It's difficult for him here. We've already got Gareth Bale, Aaron Lennon and Niko Kranjcar, who was excellent today.
"He's a class player and he's unlucky not to play more, but it's hard. He's a fantastic footballer and a great boy."
Dos Santos looked visibly irked when Defoe dived in to tap home from close range to deny him the opener in the 24th minute.
Redknapp refused to condemn Defoe for poaching Dos Santos' goal, however, admitting the former West Ham man's actions are typical of a predatory striker.
Redknapp said: "Jermain's a goalscorer, he wants to score all the time.
"He said it was going wide."
At no moment in the match did it look like npower League Two side Cheltenham were going to cause an upset against a team that started the day 67 places above them in the league ladder. Pavlyuchenko's movement allowed him the room to tap home from eight yards after good work from Defoe and Dos Santos down the left to make it 2-0 and the Mexican's skilful finish four minutes from the death highlighted the gulf in class between the two teams.
Having given his first-team stars a rest following a damaging festive period that saw Tottenham's injury list grow, Redknapp was happy to see his team progress to the fourth round without a scare.
"There were a few lads who needed a game and it was important to get a game into them," Redknapp added.
"Those games are as straightforward as you make them. They're never easy but we stuck a couple away and were always comfortable after that."
Only on a couple of occasions did Cheltenham threaten Spurs' goal.
Kaid Mohamed tested stand-in goalkeeper Carlo Cudicini early on and Russell Penn blazed over in space during the second half.
Robins boss Mark Yates, whose team are one point off League Two leaders Crawley, was happy to see his team put on a good display for the 5,000 fans who had made the trip from Gloucestershire.
"I was massively proud of the side in both halves," Yates said.
"We were never going to come here and go toe-to-toe, we tried to keep it tight like a lot of teams have and we showed we've got decent players.
"It was a reward for the hard work the players have put in and they can be really proud of it."