The Cottagers struggled for possession in the first half and went 2-0 down within 30 minutes thanks to goals from Nicola Legrottaglie and Jonathan Zebina.
Dickson Etuhu pulled one back but Roy Hodgson's men always looked second best and David Trezeguet's sharp third goal emphasised the gulf in class between the two sides.
Hodgson handed Paul Konchesky his first start in almost two months in the place of cup-tied Nicky Shorey while Clint Dempsey, who picked up a serious knee injury in January, was a surprise inclusion on the substitutes' bench.
Alberto Zaccheroni made four changes to the Juventus side that beat Fiorentina last weekend, among them World Cup winners Fabio Cannavaro and Fabio Grosso.
While the game represented arguably the most important in Fulham's history, home enthusiasm for the match was not quite on the same level.
The 28,000 capacity Stadio Olimpico di Torino only contained 11,406 people with around 1,100 of them being Fulham fans.
Hodgson admitted before kick-off his side were playing one of the biggest matches in the club's history and they soon looked out of their depth against a fluent Juve side dripping with class.
The Old Lady, currently fifth in Serie A, began with a fluid passing game and Mark Schwarzer soon found his goal under attack when he reacted well to tip Trezeguet's goal-bound strike around his left hand post.
The Australian was beaten seconds later though as Legrottaglie found his way through a packed penalty box to power a close range header past the keeper from the resulting corner.
Fulham could have levelled in the 16th minute but Simon Davies miskicked from eight yards out after an impressive run from Damien Duff down the right.
The Bianconeri continued to threaten the Cottagers' box and the front three of Antonio Candreva, Diego and Trezeguet were proving to be too much of a handful for the Fulham midfield, who struggled to get possession.
Fulham top scorer Bobby Zamora looked frustrated and isolated up front but he won the free-kick that put the Cottagers momentarily back in the tie.
Davies whipped a free-kick on the edge of the box after a foul on the former Spurs hitman and the ball fell to Etuhu who slotted the ball past an off-balance Alex Manninger via a deflection.
Sensing an equaliser, the travelling fans urged on Hodgson's men and Brede Hangeland and Zoltan Gera were guilty of firing close-range efforts straight at Manninger.
Schwarzer then had to pull off a fantastic one-handed save but Juve went 3-1 ahead when Trezeguet fired past the Fulham stopper on the stroke of half-time after his original shot smacked against the woodwork.
The second half started at a slow pace and Juventus stroked the ball about the pitch with ease.
Aaron Hughes, who was captaining the side in the absence of Danny Murphy, pulled off a crucial challenge on Trezeguet inside the box at the start of the second half as Juve looked for a fourth.
Zamora chose to plough the flanks in an effort to create chances but the two dangerous crosses he produced were thwarted by Italy centre half Legrottaglie.
Hodgson's side lacked creativity and the Fulham boss looked for fresh impetus from Dempsey, who came on for Davies to make his first appearance since hobbling off against Blackburn in January.
Trezeguet was given a rapturous reception by the home fans when he was replaced by Vincenzo Iaquinta.
A deliberate handball brought a booking for Legrottaglie and gave Fulham a free-kick from 20 yards out but Duff's curling cross was expertly cut out by Cannavaro.
Former Liverpool player Mohamed Sissoko replaced the solid Christian Poulsen in midfield with 15 minutes left.
Zamora won a corner for the Cottagers with eight minutes left but Jonathan Greening's dangerous cross was cut out by Iaquinta.
The former Manchester United man's frustration was clear as he received a booking for a needless foul on Candreva in the Juve half.
Claudio Marchisio, who was a doubt with illness, jinked inside the Fulham box but his shot was blocked.
Diego almost made it four but his 25-yard stinging strike was tipped wide by Schwarzer.
Dempsey fired Fulham's last meaningful effort wide to ensure that the first meeting between the two sides ended in a 3-1 defeat, leaving Hodgson's men with much work to do in the second leg next Thursday.