Rooney notched his fifth goal of the season and showed that United are serious contenders to retain their Premier League crown despite the loss of Cristiano Ronaldo during the summer.
Despite Jermain Defoe firing Tottenham into the lead, United hit back with goals from Ryan Giggs, Anderson and Rooney and ended Tottenham's 100 per cent start to the campaign.
Their victory was even sweeter after Paul Scholes was shown the red card in the second half for two bookable offences.
Tottenham got off to a dream start when they scored in the opening 60 seconds.
Benoit Assou-Ekotto crossed into the box. Nemanja Vidic headed the ball across goal and Defoe was on hand to fire a spectacular overhead kick past Ben Foster into the corner of the net.
But United got themselves back into the game when they drew level on 25 minutes.
Wilson Palacios was penalised for fouling Dimitar Berbatov on the edge of the box Giggs curled a left-foot free-kick past the stranded Carlo Cudicini into the roof of the net.
Berbatov almost made a dream return to his former club on 27 minutes when he beat Robbie Keane and fired a fierce shot which produced an excellent stop from Cudicini, who did well to tip the ball over the bar.
United blew a golden chance to take the led on the half hour mark. Patrice Evra crossed for Rooney who saw his shot beaten out by Cudicini. The rebound fell to Berbatov, who saw his shot cleared off the line by Sebastien Bassong.
Berbatov then wasted another golden chance when he struck the ball over the bar from close range.
But the Premier League champions managed to grab a second on 41 minutes.
Scholes' shot was only half cleared by Ledley King. The ball fell nicely to Anderson on the edge of the box and he drilled a low shot past Cudicini into the corner of the net.
Jermaine Jenas was brought on at half-time and almost made an immediate impact on 54 minutes when his fierce drive was superbly saved by Foster.
From the resulting corner, Peter Crouch rose the highest in the box and saw his header strike the top of the bar.
United were reduced to ten men on 58 minutes when Scholes was shown his second yellow card of the game following a lunging challenge on Tom Huddlestone.
Despite the loss of Scholes, United could have added a third on 67 minutes when Rooney broke clear of the Tottenham defence and saw his shot pushed onto the bar by Cudicini.
But United sealed the victory when Rooney made it three for United on 78 minutes.
Darren Fletcher's pass picked out Rooney who skipped past Alan Hutton and fired the ball through the legs of Cudicini.