The Iceland international netted one either side of half-time, but the score could have reached double figures such was the hosts' control of the game.
Tottenham created chances galore, with new signings Christian Eriksen and Paulinho particularly impressive and Norwich had Sebastian Bassong and John Ruddy to thank for keeping the scoreline respectable.
Norwich were without Ryan Bennett (knee) and long-term absentee Elliot Bennett, but close-season signing Gary Hooper, still awaiting his debut due to a shin injury, made the bench.
Andre Villas Boas, meanwhile, handed a debut to Eriksen and named club-record signing Erik Lamela among his substitutes. Emmanuel Adebayor, Etienne Capoue and Aaron Lennon all missed out through injury.
Tottenham flew out of the blocks, creating two chances within the first four minutes and setting the tone for a dominant first-half display.
The hosts were yet to score from open play this season but first Sigurdsson and then Andros Townsend almost rectified this, both missing the target with shots from just outside the area.
Eriksen then had a good chance, jinking inside from the right and shooting low to draw a good save from Ruddy. Roberto Soldado's audacious back-heel from the rebound was saved by a combination of Bassong and the post.
And ex-Tottenham defender Bassong again came to Norwich's rescue minutes later. Moussa Dembele took out two defenders with a feint and cut-back to Soldado, but the Cameroon international blocked his goal-bound effort.
A Tottenham goal looked inevitable, and so it proved in the 28th minute when Eriksen laid the ball into Sigurdsson's path as he burst into the area and the former Hoffenheim man finished coolly past Ruddy.
Norwich had not been in the game, but they had a half-chance 34 minutes in through Ricky van Wolfswinkel. The Dutchman met Nathan Redmond's left-wing cross but was thwarted by a last-ditch Danny Rose tackle.
Rose was involved at the other end shortly before the interval, but his first-time strike at the back post soared well over.
Tottenham picked up where they left off after half-time and were soon 2-0 up. Paulinho’s low cross from the right evaded Ruddy’s dive, presenting Sigurdsson with an open goal.
The midfielder came close to getting his hat-trick on the hour mark, but Ruddy was equal to his rasping 20-yard drive.
Norwich were buckling under concerted Tottenham pressure and they had Michael Turner to thank for not conceding a third, as he made a fantastic intervention to deflect Townsend’s right-wing cross over the bar, with Soldado lurking behind him.
Chris Hughton’s side finally had a chance of their own on 68 minutes, the ball breaking to Van Wolfswinkel in the area, but Rose foiled the former Sporting Lisbon striker with a good block.
Eriksen made way for Lewis Holtby in the 71st minute, bringing a standing ovation from the White Hart Lane crowd in appreciation of an impressive debut from the ex-Ajax playmaker.
There was controversy late on as Hugo Lloris left his area to make a punched clearance. The Norwich fans called for a red card but referee Lee Mason was lenient, only showing a yellow to the France international.
