Mjallby took to European competition like ducks to water when beating Lincoln Red Imps in the previous round, but the Slovak champions were always expected to provide a tougher test.
So it was proving in the first half, as only some last-ditch defending prevented Mykola Kukharevych and Alasana Yirajang giving Slovan Bratislava the lead, before the visitors went close again early in the second half, only for Robin Wallinder to deny Sandro Cruz's sweet strike.
Having soaked up a lot of pressure, Mjallby grew into the game - Jeppe Kjaer weaved through the defence before forcing Dominik Takac into a fine stop, before Kenan Bajric handled inside his own area to give the hosts the chance to take the lead from the penalty spot.
Elliot Stroud's run up was short, but it didn't matter, as he sent Takac the wrong way to put MAIF ahead with an hour gone.
But things turned sour for Mjallby with 12 minutes to go, as Kukharevych played the ball through to Suleiman Camara, who picked his spot from the edge of the box, striking the ball across Wallinder and into the bottom corner.
Karl Marius Aksum's side looked resigned to heading to Slovakia level, but the visitors had other ideas. In the 94th minute, Camara turned provider, squaring across the area to Manasse Kianga, who held his nerve to jink past the defender before rolling the ball into the corner to win it for Slovan.
It's a first European defeat for Mjallby, who must now get a victory in 90 minutes when the two meet again on 11th August.