Three first-half goals from the Italian giants have left them firm favourites to progress following the second leg in London next week, but the Cottagers have been given hope of an upset after a hard-working performance left them with a foothold in the tie.
Juventus drew first blood when Nicola Legrottaglie headed them into an eighth-minute lead.
Mark Schwarzer conceded a corner when he turned David Trezeguet's shot around the post and Legrottaglie made the most of the set-piece as he rose highest to power home a header from eight yards.
Simon Davies fluffed his lines soon after when he took a swipe at fresh air from 10 yards out and the Cottages were left to rue his profligacy when they fell 2-0 behind on 25 minutes.
Jonathan Zebina waltzed past Davies and Dickson Etuhu before letting fly with a 25-yard shot which arrowed into the far corner of the net.
Juve's front three of Trezequet, Diego and Antonio Candreva were looking dangerous every time the ball entered Fulham's half and the less than half full Stadio Olimpico di Torino were enjoying the performance.
However, the 1,500 travelling fans were brought to the life in the 36th minute when Etuhu halved the deficit following a lucky deflection.
Davies' free-kick was only half-cleared to Etuhu, who let fly with a powerful shot which was heading wide until it took a major deflection off Legrottaglie and left goalkeeper Alex Manninger completely helpless.
Fulham came close to levelling the game when Manninger pulled off a great save to deny Brede Hangeland's close-range header soon after.
At the other end, Schwarzer had to be at his best top tip over a thunderous drive from Fabio Grosso.
But any hopes Fulham had of battening down the hatches having secured an away goal were ended on the stroke of half-time when Juve restored their two-goal advantage.
Trezeguet's shot rattled the post but the ball rebounded back to the France striker and he prodded it home with the help of the body of team-mate Salihamidzic.
Juve looked content to retain possession at the start of the second half as they probed patiently for a fourth goal.
Following 15 minutes without a clear goalscoring opportunity, Fulham boss Roy Hodgson sent on Clint Dempsey for his first appearance since suffering a knee injury in January.
Vincenzo Iaquinta came on for Trezeguet for the home side as they continued to hold the upper hand without threatening to go for the jugular.
Fulham dominated possession in the closing stages but could not find a way to open up a well-drilled defensive unit.
Meanwhile, Diego's last minute 20-yard shot, which Schwarzer palmed away at full-stretch, highlighted how dangerous Juve will be at Craven Cottage in the second leg next week.