Boss Hugh Kelly threw in new full-back signing Mark McGill for a daunting debut against the West Region big guns.
But the 23-year-old put the ball into his own net for what proved to be the game's only goal with 20 minutes left.
And Kelly's surprise ruse to spike the Ayrshire outfit's intricate passing style by narrowing the Greenfield Park playing surface by four yards on each side was rumbled when the fourth-round replay was postponed last weekend and Medda officials present at the pitch inspection reported back to gaffer Chris Strain.
He made the switch to team up towering pair Mark Campbell and Willie Kinniburgh in central defence and they proved instrumental in the Ayrshire side's clean sheet.
Kelly said: "Mark is the grandson of former Shettleston manager and legendary Third Lanark winger Joe McInness, who is now in his eighties, but who still takes in Junior games.
"He introduced the lad a few weeks ago and he has been training alongside the other guys, but in a kickabout last week he really caught the eye and I signed him thinking he'd strengthen us defensively.
"Anyone watching the game today would agree he did that and more, and I felt he was nothing short of immense against a very experienced Medda left flank. He is in no way accountable for their winning goal today.
"His was the last touch, but he had no choice but to go for the ball as it was a certain goal anyway with Keigan Parker standing behind him on the goal-line ready to knock the ball into the empty net.
"Mark can be proud of his display, as can every Shettleston player today, because you would never have known two full leagues separate the clubs.
"It wasn't a great footballing spectacle, but we set our stall out to win ugly and we were the better side in the first half and should have scored early doors. If we had, I'm pretty sure we would be the ones preparing to meet Cumnock in the next round."
The home side were lively from the kick-off and passed up a glorious scoring opportunity when an unmarked Danny Boyle headed over the crossbar.
Narrowing the pitch certainly ensured Colin Crichton's long throws were a big threat, however Medda centreback Campbell won just about every aerial challenge and what he did miss did not escape defensive partner Kinninburgh. After weathering an early onslaught, Irvine Meadow then came close to an opening goal when Mark Mclennan's first time 22 yard shot hit the top of the bar.
The stalemate continued after the break, and looked like it might even last to the final whistle until Medda took the lead on 70 minutes.
Town centre-back Ross Dickson was caught under a high Zander Ryan clearance and sent a sclaffed header into the path of Richie Barr, whose fiercely driven cross-goal ball went in off the unlucky McGill.
The debutant said: "Nobody is blaming me, but it's hard not to feel gutted at how things have panned out in my first game for the club.
"Their forward was waiting to apply the finishing touch so I threw my body in, but went with my wrong foot and could do nothing but help the ball over the line.
Happy Meadow boss Strain emerged to say, "That wasn't football as I know it. My half-time team talk consisted of me telling players not to make any passes."