Jose Mourinho's side took control early on as a Lucas Moura strike was followed by an Andres Andrade own goal, with substitute Son Heung-min adding a third goal late on.
Spurs' starting line-up featured two eye-catching names, with recent signings Carlos Vinicius and Gareth Bale included.
But, while the first 10 minutes saw Bale evidencing his threat from a starting position on the right wing, they also featured a poor Vinicius miss, the striker heading wide on his full debut when he should have done better with a teasing Matt Doherty cross.
He did, though, waste little time in making up for that error, latching onto a Ben Davies ball over the top soon after to tee up compatriot Lucas for a composed finish.
Bale, too, had his moment before the half-hour mark, forcing Andrade to put through his own net with a low, outside-of-the-boot cross at the end of a sweeping move started by Sergio Reguilon's brilliant surge forward.
At that point it looked like Spurs might run away with it, but LASK rallied to enjoy a strong finish to the half, with Andreas Gruber coming closest to making it count courtesy of a fine drive from distance.
They were also not far away form reducing the deficit on the hour mark, Petar Filipovic's header from a corner sailing just over the bar with Joe Hart and his defenders beaten.
However, with the Austrians enjoying their best spell of the game, Mourinho was able to call for the cavalry in the form of Moussa Sissoko, Dele Alli and Son.
It was the latter pair who made the most telling difference, with Alli forcing a smart save after being played through by Vinicius after Son firing over the bar from close range.
Spurs never looked like rueing those misses, however, and had time to cap their win with a third goal courtesy of a side-footed Son from close range.
What does it mean? Spurs' show of strength
Although three points will have been Mourinho's main concern, he was also given welcome evidence of Spurs' new-found strength in depth.
Talismanic striker Harry Kane was barely missed, while the hosts were able to bring the likes of Son and Alli off the bench late on.
Doherty gave another demonstration of why Spurs lured him from Premier League rivals Wolves.
The Irish full-back took a team-leading 104 touches, provided two key passes, made four tackles and completed over 80 per cent of his passes in the opposition half.
More to come from Bale
Although he played a key role in Tottenham's second goal on his first start since returning, Bale should have much more to offer once he builds up match fitness in the coming weeks.
In his hour on the pitch, the Welshman took just 44 touches, and failed to produce a shot or a key pass while losing possession on 20 occasions.
Tottenham must wait until Monday for their next Premier League outing against Burnley at Turf Moor, with their European campaign continuing away to Antwerp next Thursday.