A 1-0 victory over Chelsea lifted the Merseyside club into the top half of the Premier League table and will ease the early pressure on the former Wigan boss after three games without a win at the start of the season.
And Saturday Night Football expert Redknapp said that, while Martinez is trying to encourage a more possession-dependent style of play, the determined style of Moyes' Everton is still there for all to see.
"To beat a top side like that will mean so much to the team, the ethic, the dressing room and the manager," he said.
"People have been questioning the philosophy that he's trying to play football with, but what's apparent today is that they still have the Moyes intensity about the way they play their game.
"I thought they were magnificent today because it was a tough game. In the second half they had to hang on and there were times when it was backs against the wall with crosses coming into the box.
"But I thought they were magnificent today and still had an attacking threat. Every time the ball went forward you thought somebody could do something.
"That will bring a lot of satisfaction to that dressing room."
Redknapp said it will be tough for Martinez to fill the shoes of Moyes, who departed after a successful decade at Goodison Park to take over at Old Trafford.
But he insists that it won't be too difficult to impose a fluent style of football because Everton have played attractively for several years.
Redknapp added: "There's always pressure when you're a manager and you haven't won, especially when you're following somebody like David Moyes, who has been so successful, and you're also trying to do things differently.
"We've got to be a bit careful. I've watched Everton for the last few years and they play good football, they've got good players and had three England internationals in their team today.
"They weren't a long ball team, we have to get that straight, and last year they had such a good season.
"They showed determination and the fans were right behind them as well. There were times when you could see they wanted to get balls into the box, but they were trying to play.
"I like the way that Martinez teams play. He did a great job at Wigan and I've got no doubt he'll do a really good job at Everton."