Ivelin Popov's second-half goals were the difference between the two sides at Fir Park and he twice finished off passes from dangerman Ibrahima Balde to give the visitors a perfect introduction to European football.
Motherwell had responded well to Popov's opener but they were cut open on the break in the 77th minute of the third qualifying round first leg.
Motherwell lost 2-0 at home to Panathinaikos and Levante last season to see
their UEFA Champions League and Europa League campaigns effectively ended before the second leg.
In both ties Motherwell had plenty of possession but were undone by clinical
opponents who both scored their killer goals on the break after good spells of
home pressure, and manager Stuart McCall was determined not to suffer the same fate again.
To that end, he named a more cautious line-up with John Sutton the lone striker and left-back Zaine Francis Angol on the flank of a five-man midfield.
The game plan looked to be working reasonably well and Sutton had a first-half penalty appeal turned down, but it was the class of Ibrahima that undid last season's Scottish Premier League runners-up.
The Senegal striker's pace, movement and close control proved too much and he could have had a first-half hat-trick before his penalty-box trickery allowed
Popov to net in the 52nd minute.
Motherwell hit back with a sustained spell of pressure but last-man Stevie
Hammell gave the ball away after Kuban cleared a corner and the hosts were
punished with a swift and brutal counter-attack.
There was more misery for Motherwell in stoppage time when Sutton headed
against the post from Francis-Angol's cross.