Gonzalo Higuain and Paulo Dybala each scored in the space of 2 minutes and 45 seconds in the second half to send the Serie A champion into the last eight.
Spurs became the first team ever to overturn a two-goal deficit at Allianz Stadium in the first leg and picked up where they left off at Wembley, with Son Heung-min giving them a deserved half-time lead.
However, a change in system and personnel from Massimiliano Allegri turned the match on its head, with Higuain levelling the match before setting up Dybala to score his first UEFA Champions League goal in 330 days.
Mauricio Pochettino's side pushed hard in the closing stages but it could not find a goal to send the tie into extra time, as Juve became the first team to beat Spurs in the competition this season and secure a spot in the last eight.
Spurs kept Juve pegged back in the opening exchanges but were dealt a huge slice of luck by referee Szymon Marciniak 17 minutes in, as Jan Vertonghen escaped punishment for a clumsy challenge on Douglas Costa in the penalty area.
Buffon was certainly the busier of the goalkeepers, though, the veteran making two good saves from Son and one smart stop from Dele Alli's volley, while Juve found it difficult to bring the front two of Higuain and Dybala into the action.
There was another controversial incident that escaped the referee's notice half an hour in, with Andrea Barzagli standing on Son before flicking a boot in his direction as the Korean lay on the ground.
Son had the last laugh six minutes before the break, though, scuffing a shot over Giorgio Chiellini and Buffon and into the net after being picked out by Kieran Trippier's cross.
Allegri made a tactical switch 60 minutes in, bringing on Stephan Lichtsteiner and Kwadwo Asamoah, and the latter almost had an immediate impact when his cross was turned into Dybala's path, but the Argentina forward blazed wide.
The change did prove telling four minutes later. Lichtsteiner escaped down the right and clipped over a cross, Sami Khedira nodded the ball towards goal, and Higuain prodded it past Hugo Lloris and into the net.
It was Juve's first shot on target and it duly scored with its second minutes later. Higuain drew Davinson Sanchez out of position and played a fine reverse pass to Dybala, who raced through on Lloris's goal and fired the ball into the top-right corner.
Spurs regrouped and pushed forward in search of an equaliser, with Chiellini making a vital interception right in front of goal and Son shooting inches wide of the left-hand post.
Harry Kane had made little impact but he almost snatched a 90th-minute equaliser, his header coming back off the post before being hooked off the line by Barzagli, as Juve just held on to keep its treble pursuit alive.