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 involved.
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 from 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 match, 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 victory with a third goal thanks to a side-footed Son from close range.
Tottenham must wait until Monday for their next Premier League outing versus Burnley at Turf Moor, with their European campaign continuing away to Antwerp next Thursday.
