Andy Johnson put the hosts ahead on 19 minutes, with highly-rated Netherlands striker Luuk de Jong on target five minutes before the break.
Both sides tried to win the match, with Fulham creating the better chances and the Eredivisie runners-up dangerous on the break, but neither were able to find a route past the impressive goalkeepers and a draw was the fair result.
It seems strange to say but Fulham and Twente started Group K as favourites - OB and Wisla Krakow are not expected to qualify - so it was a decent point for both sides.
The Dutch side, who failed to qualify for the Champions League after a play-off defeat to Benfica, started as well as one would expect, holding on to the ball well and creating an early chance as Ola John - brother of former Fulham forward Collins - fired weakly at Mark Schwarzer after a mazy run.
Fulham found their groove though, stringing long stretches of passing together and pegging the visitors back for 20 minutes or so.
In that time Bulgaria goalkeeper Nikolay Mihaylov made a smart stop from Clint Dempsey’s rising drive, Pajtim Kasami slipped just as he was set to finish off a sweeping move, while Danny Murphy put wide when Kasami’s high ball bamboozled hapless full-back Dwight Tiendalli.
The Whites also scored in that period, thanks to a fine finish from Johnson, who rounded the keeper and clipped the ball home from an acute angle after latching on to a suicidal pass across goal by Tiendalli.
But just after the half-hour mark Twente found their mojo, as the formerly quiet Roberto Rosales swung in a brilliant cross that De Jong could only head at the well-placed Schwarzer.
That appeared to return a spring to the Dutch side’s step and they bossed the remainder of the half, drawing level when De Jong again met a Roberto Rosales cross, helped by some slack marking as he picked his spot this time, too far for Schwarzer to reach.
It was a cracking first half and the second period continued in the same vein as De Jong narrowly missed out when his wonderful run split the Fulham defence, only for Tim Cornelisse’s pass to be slightly overhit.
Fulham realised they could well be stung by Twente and responded in superb style: led by Murphy and bounding full-back Matt Briggs, they took the game by the scruff of the neck, dominating possession and creating chance after chance.
First Dempsey was unfortunate when his brave diving header flew wide with Mihaylov beaten, while the Bulgarian made two super stops from Johnson, the first at his feet and the second from close range after Dempsey slipped the striker in.
Johnson thought he should have had a penalty from an incident earlier in that latter move, and he was having his best game for some time, doing exceptionally well down the left before cutting it back to the sliding Steve Sidwell, who was denied by Douglas.
Mihaylov - who was faultless but repeatedly beaten in matches against English opposition while playing for Bulgaria - was having a blinder, and he made another smart stop from Dempsey after he again combined with Johnson.
Meanwhile Sidwell missed a good opportunity, heading over after escaping the attentions of Douglas, as Fulham seemed unable to capitalise on their territorial dominance.
And then, as with the first half, a spell of Twente superiority followed, with Fulham looking a touch flaky defensively as Brede Hangeland made an uncharacteristic error that Rosales and De Jong were unable to fully exploit as they both slipped at vital moments.
Briggs - who had impressed greatly until he was left labouring following a foul for which Rosales was booked - had started to make mistakes and was promptly brought off for Philippe Senderos.
Damien Duff was also brought on as Fulham went for the win, having resisted that Twente surge and finishing the game very well.
But it was the visitors who had the final chance as the ball fell to Rosales on the edge of the box, but the Venezuela winger was only able to drive wide with the goal gaping.
It would have been harsh on Martin Jol’s side, but a creditable point against fancied opposition is a decent enough way to start the European season.