More than two years after Anderson joined United from FC Porto, the 21-year-old Brazil midfielder's powerful 41st-minute strike consigned Spurs to their first loss of the season.
"As a young player he was always capable of scoring goals," manager Alex Ferguson. "Hopefully that's the encouragement that will see him do more of that. He had a marvelous performance for us today."
Jermain Defoe's early acrobatic overhead kick had stunned the visitors, but Ryan Giggs leveled in the 25th when he curled in a free kick.
Despite the Red Devils playing more than 30 minutes with 10 men after Paul Scholes was sent off, Wayne Rooney was able to add a third in the 78th from an energetic counterattack.
"Rooney works his socks off, holds the ball up, dribbles and beats people, scores goals, heads it, tackles you twice as hard as anyone else and doesn't stop running around," Spurs manager Harry Redknapp said. "He's like a little kid in the playground."
The victory moved Ferguson's side up to second in the standings, above Manchester City and Spurs on goal difference. Chelsea has a three-point lead after opening the season with five wins.
Spurs had been enjoying their best start to a season since last winning the title in 1960-61 and their vibrancy took just 52 seconds to unpick United.
Benoit Assou-Ekotto scooped the ball over to the far post and Nemanja Vidic headed it into the center while trying to outmuscle Peter Crouch.
The ball dropped for Defoe and the England striker beat goalkeeper Ben Foster with a spectacular overhead kick.
Former Spurs striker Dimitar Berbatov was denied an equalizer when Carlo Cudicini was equal to the Bulgarian's glancing header from Ryan Giggs' inswinging free kick.
But with his next set piece, the 35-year-old Giggs -- who was captain for the day on his 700th start -- curled the ball into the top corner to maintain his record as the only player to score every season since the Premier League's inception in 1992-93.
"We didn't have a look in with free kicks when Cristiano was here," Giggs said, referring to Cristiano Ronaldo's offseason transfer to Real Madrid.
After such a bright start, Spurs were now feeling the power of United. A shot from Berbatov was pushed over by Cudicini before United came close in quick succession in the 30th.
First Rooney was denied by Cudicini's block, then Berbatov's effort on the rebound was cleared off the line by Sebastien Bassong and the Bulgaria striker struck over when he pounced again.
Tottenham's main threat came in the air from the lanky Crouch, who headed just wide when he rose above Rio Ferdinand to meet a corner.
The England forward was making his first league start since an offseason move from Portsmouth, replacing the injured Luka Modric to partner Defoe while Robbie Keane was on the left.
"We missed Modric because he makes us play -- he is a fantastic player and makes us tick," Redknapp said. "It's difficult to get (Keane, Defoe and Crouch) in the team together. It is a problem getting the balance and I'll have to work it out this week."
There was no way back for Tottenham after Anderson's powerful, low strike that slipped through Cudicini's fingers.
After the break, it was Foster's left fingertips that denied Jermaine Jenas as the midfielder made his first appearance of the season following a calf injury. From the resulting corner, Crouch headed against the crossbar.
United was reduced to 10 men in the 59th when Scholes was shown a second yellow card for clattering into Tom Huddlestone. The 34-year-old midfielder, who was sent off twice last season, had been booked at the start of the half for a badly timed challenge on Defoe.
"It was a terrible sending off, I think the referee got it wrong," Ferguson said. "I've looked at the video twice and I don't see any sending off. I think he was sent off because he is Paul Scholes."
Ferguson responded by replacing Berbatov with midfielder Michael Carrick, another former Spurs player.
But a superb solo effort from Rooney on the counterattack sealed the points as the striker collected Darren Fletcher's ball and cut inside Alan Hutton and shot through Cudicini's legs.