Harry Kewell tries to find a little space in midfield
Leeds United's European rollercoaster ride of the past three seasons took a stomach-churning dip on the holiday island of Madeira.
David O'Leary's side reached the Uefa Cup semi-final two seasons ago and progressed to the same stage in the Champions' League last term to leave them among the favourites for this competition.
Leeds' sluggish start was punished on 34 minutes, though there appeared little danger when Rio Ferdinand was penalised for a foul on Quim 40 yards out.
Bruno Fernandes floated in a hopeful free kick and defender Dominic Matteo ducked under the ball under a challenge from Mitchell van der Gaag, leaving Nigel Martyn groping helplessly at thin air as the ball drifted into the bottom right-hand coner.
Leeds were still trying to come to terms with falling behind when their defence yawned again.
Mills waited in vain for Martyn to come off his line and collect, allowing Quim the chance to steal in but blaze a shot over the crossbar.
Leeds' frustrations continued in the second half, as the heady Champions League nights of last season seemed distant memories.
A fine one-handed save from Martyn prevented Quim from claiming a long-range second and Dinda shot straight at the keeper after Harte had presented him with an opening.
At the other end, Leeds created precious few chances.
Lee Bowyer was conspicuous by his absence and the closest Leeds came to a goal was in injury time when Gary Kelly, playing as a winger, forced a fine block from the alert Nelson.
Nelson had only previously been tested by a looping header from Mills and a forward run by Mark Viduka, who had already been flagged back for offside.
The only silver lining on a cloudy night for Leeds was that the final scoreline leaves them well in contention to turn things around in the second leg.
But, if that is to happen, a major improvement is needed from O'Leary's jaded stars.
Maritimo: Nelson, Albertino, Sergio, Zeca, Dinda, Andre., Bruno, Quim, Briguel, Kenedy., van der Gaag. Subs: Gilmar, Gaucho, Lino, Alan, Shannon, Arriola, Santos.
Leeds: Martyn, Kelly, Matteo, Ferdinand, Harte, Mills, Batty, McPhail, Kewell, Keane, Viduka. Subs: Robinson, Wilcox, Duberry, Burns, Maybury, Woodgate.
Referee: Jack Van Hulten (Holland).