In their first-ever competitive European assignment, Steve Bruce's Tigers were presented with a golden opportunity to snatch an away goal when Shane Long was brought down in the box on the hour mark at the Stadion pod Dubnom in Zilina.
But Huddlestone's resulting penalty was saved by goalkeeper Milos Volesak, and the midfielder could have scored from the rebound, only to see his follow-up effort clip the top of the crossbar on its way over.
The sides will now meet again for the second leg at the KC Stadium next Thursday as they battle to reach the playoff round.
Bruce selected a familiar starting XI for his side's first taste of European football, with summer signings Robert Snodgrass and Tom Ince on the bench.
Injury doubts Allan McGregor, Curtis Davies and David Meyler all started, with Sone Aluko partnering Long in attack and Nikica Jelavic among the substitutes.
Liam Rosenior and Ahmed Elmohamady were selected in the wide positions of a 3-5-2, leaving the £15million pair of Snodgrass and Ince to bide their time for a debut.
The FA Cup runners-up almost got off to a perfect start inside two minutes when Meyler won a free-kick and went close to converting Huddlestone's delivery at the back post.
The Tigers saw more of the ball in the opening half-hour but were kept at bay with relative comfort.
Their favoured route appeared to be crosses into the box and Meyler missed a half-chance before Long had a better opening laid on by the busy Rosenior. The Irishman was in a good position but could not keep his header down.
The momentum shifted in the 15 minutes before the break, Moses Simon and Patrik Misak beginning to threaten, but Hull's defence held out.
Alex Bruce, in particular, needed to be alert to dispossess Simon as he dribbled into the box, while McGregor was left untroubled by Haris Hajradinovic's left-footed effort.
Trencin upped the ante eight minutes into the second half, Misak going desperately close with a long-range shot that just missed the target.
Long then made what should have been a decisive contribution, cutting inside the box only to be fouled by Peter Cogley.
Huddlestone's spot-kick was saved by Volesak but the rebound fell perfectly, presenting a straighforward finish for the former Spurs man.
But Huddlestone - who missed in the shootout when Spurs lost to Basel in the 2013 competition - somehow contrived to flash his shot against the crossbar and high into the stands.
That moment knocked the wind out of Hull, with Bruce sending on his prize summer signings Ince and Snodgrass in response.
They could not change the result, but City will nevertheless head to the KC Stadium next week as favourites to reach the play-offs.