The 32-year-old United veteran, named as the Barclays Premiership player of the month for August, scored only three league goals last season.
But his ninth minute header, which ensured United their best ever start to a Premiership campaign, put him well on the way to eclipsing last season's meagre tally.
Spurs will feel they did enough to take a point from their latest outing to Old Trafford as they had done on their two previous visits.
They had their chances, but were foiled on three occasions by ultra-dependable United goalkeeper Edwin Van der Sar.
The all-important goal came in the ninth minute. Cristiano Ronaldo had tested Spurs goalkeeper Paul Robinson three minutes before with a curling shot from the right which he saved well.
But England's Number One will have been disappointed that he did not stop the goal.
Edgar Davids fouled Louis Saha 35 yards out and, although Ronaldo's right-foot free-kick was a blistering effort, it was straight at Robinson.
As the England man parried the shot and Giggs reacted first to score with a header from eight yards which went in off the underside of the bar.
United, with Michael Carrick making his home debut following a £18.6 million summer move from Spurs, failed to add a second goal.
Spurs came back at them and Jermaine Jenas who should have done better than shoot wide from 12 yards after Rio Ferdinand's mistake put United under pressure.
Van de Sar came to United's rescue in the 38th minute with a superb double save.
Davids' glorious pass found Michael Dawson racing unmarked on the right side of the United box.
Van de Sar raced off his line to block Dawson's point-blank effort and the Dutchman then put his body in the way of Didier Zokora's follow-up shot.
Kieran Richardson failed to tuck away Saha's cross early in the second half, but Spurs had an even better chance in the 52nd minute.
Jeanas set up Spurs debutant Pascal Chimbonda as he raced into the United penalty area from the right, and his shot took a deflection off Ronaldo which forced a fine save from Van der Sar.
United could have added a second goal in the 65th minute when Gary Neville found Saha on the right. He knocked the ball inside to Ronaldo, whose fierce shot from 12 yards was well saved by Robinson.
Spurs boss Martin Jol brought on Jermain Defoe for Robbie Keane and he should have scored when he ghosted in on the blind side of the United defence to meet Benoit Assou-Ekotto's cross from the left. But he placed his free header wide.
Mido had another chance when United's defence was again caught cold when he met Hossam Ghaly's cross from the left, only to head weakly at Van der Sar.
United should have put the game beyond doubt three minutes from time.
A long ball out of defence sent Saha racing clear. The Frenchman held off Assou-Ekotto's lunging tackle and rounded Robinson, only to see the Spurs goalkeeper recover and beat the effort away.