The Cottagers had gone ahead early on when Zoltan Gera's low shot crept in.
The Cottagers won 1-0 at Wolfsburg to seal a 3-1 aggregate Europa League win.
The Italians dominated but Dickson Etuhu's away goal gives the Cottagers hope of overturning the deficit.
But the professionalism of the Cottagers' display ensured that one goal was all that was needed.
Adams started at Fulham in August 1996, leading the Cottagers to promotion from Division Three in his first season.
Villa took the lead through a John Pantsil own goal before Andrew Johnson stooped to head the Cottagers level.
But the Cottagers, whose line-up featured six changes from the one that started at Burnley last weekend, were worthy victors.
The Cottagers failed in an attempt to have Pascal Zuberbuhler registered for the match, which Fulham won 1-0, after an injury crisis at the club.
Barring a spell in the first half, during which Etuhu claimed his goal, the Cottagers were second best against a strong Juve side.
After failing to make home advantage tell in Thursday's first-leg 0-0 draw against the Cottagers, Hamburg were drubbed 5-1 at Hoffenheim on Sunday.
Hodgson's men were working extremely hard to contain the slick-passing hosts, but it was actually the Cottagers who came closest to taking the lead.
That secured European football for the Cottagers and led to Hodgson being named on the shortlist for the Premier League manager of the year award.
It is a remarkable achievement for the Cottagers, who in 1996 were eight places off the bottom of the entire Football League in League Two.
The Cottagers, who protected their unbeaten league record, upped the ante after the break and Everton keeper Tim Howard was drawn more into the action.
The Cottagers had Erik Nevland sent off by referee Kevin Blom when 1-0 up and went 2-1 down before Paul Konchesky was also dismissed late on.
For Fulham and their fans, it was the perfect start as the Cottagers raced out of the blocks to stun Shakhtar and take an early lead.
And Fulham eventually buckled when John Arne Riise's shot took deflections off Gera and his brother Bjorn on the way past wrong-footed Cottagers keeper Mark Schwarzer.
The Cottagers have eased any relegation fears by taking 11 points from a possible 18 and twice came back from a goal down to earn a point.
The Cottagers justified the 65 league places that separate the sides with a professional and efficient display that ensured their progress to the last eight of the competition.
The Cottagers took 21 seconds to make amends for Wolfsburg's late goal in the first leg as Bobby Zamora neatly turned Jan Simunek and slotted his side ahead.
The Cottagers were visibly deflated by the immediate reply and Huddlestone compounded their misery with a fine fourth, driving into the corner right-footed from the edge of the area.
It is all a far cry from the final day of the 2007-08 Premier League season, when Murphy's 76th-minute winner against Portsmouth saved the Cottagers from relegation on goal difference alone.
Hull put in a resilient performance against a Fulham side which failed to take their chances and Brown says the victory had put the 10th-placed Cottagers back in the survival fight.
Meanwhile, Fulham boss Roy Hodgson, who has been mentioned as a potential successor for Benitez after guiding the Cottagers to the Europa League final, dismissed speculation linking him with the Anfield hotseat.
They could have gone in front inside 15 seconds when Brede Hangeland's header was superbly tipped over by Andriy Pyatov, but the Cottagers only had to wait two more minutes to nudge ahead.
Brede Hangeland nodded a corner wide, Zamora shot straight at keeper Povilas Valincius and Zoltan Gera flashed wide of the near post as the Cottagers failed to turn their increasing dominance into goals.
Suddenly the Cottagers were rampant and they went close to levelling soon after, when Alex Manninger clawed away a Brede Hangeland header, then threatened again with shots from Zoltan Gera and Bobby Zamora.
The Cottagers had a level of control over the game and Bobby Zamora set up Dempsey at the far post but his shot was blocked by Ian Ashbee with keeper Duke out of position.
But the injury to Zamora, who had only just recovered from a thigh injury and was playing his first game since signing a new four-year deal with the Cottagers, will leave them with mixed emotions.
The Cottagers, who will be without suspended right-back Chris Baird for the second leg, must be admired for the way they held firm to the end, Schwarzer thwarting an angled effort from substitute Mladen Petric on 88 minutes.
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