Stuart Jones capitalised on some poor defending midway through the first half to head home a famous winner for Llanelli at Airdrie's Excelsior Stadium.
Gannon, who was unveiled as Motherwell manager on Tuesday, had called on his new players to implement some of the ideas he had introduced in his first few training sessions.
But they lacked any invention as Motherwell struggled to break down the organised Welsh Premier League runners-up.
The shock factor in the result was dissipated to some extent by the under-strength nature of Gannon's squad.
With Cardiff-bound Paul Quinn about to become the ninth first-team player to leave Fir Park this summer, Gannon handed debuts to former Inverness goalkeeper Michael Fraser and young midfielder Ross Forbes. Teenager Steven Saunders came in at right back to face up to 42-year-old Llanelli player-manager Andy Legg.
There was little evidence in the opening minutes that Motherwell would struggle as they created two clear chances.
Jamie Murphy beat the offside trap to run on to a lofted pass from Forbes but he elected to square for John Sutton when a shot was on and Llanelli cleared. Saunders then volleyed over from 10 yards from Stevie Hammell's corner.
Motherwell were dominating possession but most of it was sideways and in front of Llanelli's well-organised defensive set-up.
The opener still came as a shock though as Motherwell paid the price for some static defending in the 28th minute. Legg swung in a corner to the back post and Jones headed past the stranded Fraser.
The Welsh side put themselves about and Jones, Rhys Griffiths and Michael Howard were all booked for blatant fouls in the first period.
Murphy and Jim O'Brien, playing on the wing in Gannon's 4-2-3-1 formation, struggled to get round the back of the Llanelli defence and Motherwell's only half-chances before the break came from crosses from deep positions.
John Sutton, Saunders and McGarry all headed harmlessly wide and Ashley Morris comfortably made his first save in injury-time from McGarry's header.
Teenage winger Paul Slane came on at half-time for Lasley, who had been on the end of a particularly bad challenge from Griffiths, and almost made an immediate impact.
Slane beat three men only for Thomas to head his shot away from the goalmouth.
Sutton just failed to connect with McGarry's header across the face of goal in the 55th minute, shortly before his header from O'Brien's cross forced another comfortable catch from Morris. Motherwell were restricted to long-range shots after that and the only time they came close was in the final minute.
Slane fired towards the bottom corner from 25 yards but Morris dived to push wide.
Although Gannon signed Steven Jennings from Tranmere before the game, he was ineligible to play and the former Stockport boss is restricted to the same squad for next Thursday's second leg.
They will need to find some inspiration if they are to meet Albanian side Flamurtari in the next round.