Moyes took charge of the club back in 2002 and is regarded as one of the best young managers in the game following his impressive spell at Goodison Park.
Everton have continued to challenge in the top half of the table and this latest victory means Moyes' team have lost just once in their last seven games.
In contrast, Redknapp knows he still faces a big task to pull Tottenham away from the foot of the table. Tottenham may have won seven of their last nine games, but the former Portsmouth boss knows he will have to make White Hart Lane a fortress again if they are to avoid a relegation dogfight.
Tom Huddlestone almost gifted Everton the lead on 33 minutes when he lost control of the ball and allowed Marouane Fellaini a free run at goal.
Fellaini had all the time in the world to pick out one of his team-mates inside the box, but his cross was superbly kept out by Heurelho Gomes.
Tottenham raced up the other end of the pitch and Didier Zokora released Aaron Lennon who was completely unmarked inside the box but he fired a right-foot shot over the bar.
Gomes has come under fire for a number of key mistakes this season, but he made a superb double save to keep his side in the game on 38 minutes.
Fellaini saw his fierce drive kept out by the Brazilian keeper andm when the rebound fell to the Everton midfielder, he saw the Tottenham keeper make another fine stop.
Everton continued to mount pressure on the Tottenham goal and had a golden chance to take the lead on the stroke of half-time.
Tim Cahill's cross picked out substitute Louis Saha who had a free header inside the box but he flashed his close-range effort straight at Gomes.
Everton finally got their just rewards when they opened the scoring on 51 minutes. Mikel Arteta's quickly taken free-kick picked out Pienaar whose shot took a wicked deflection off Vedran Corluka and flew past Gomes.
Roman Pavlyuchenko could have grabbed an equaliser for Tottenham on 72 minutes when his low show was superbly kept out by Tim Howard, but Everton held on to record a crucial away win at White Hart Lane.