The Steelmen were handed a place in the qualifying rounds thanks to their second place in the Scottish Premier League's Fair Play League, after Scotland's impressive disciplinary record had secured the country an extra place in Europe.
And the team who finished seventh in last season's SPL had hoped to return from Romania with a respectable result, to give themselves every chance in the second leg at Airdrie's Shyberry Excelsior Stadium next Thursday.
But an own goal from Stephen Craigan was followed by further goals from Banel Nicolita and Bogdan Stancu, and Jim Gannon's visitors could not respond.
On-loan Everton goalkeeper John Ruddy and former Northampton midfielder Giles Coke both made their debuts but Motherwell were slow to start and their evening went from bad to worse.
Bogdan Stancu forced Ruddy to make an early save, and Romeo Surdu also went close from long range for the home side.
A goal was coming and it arrived after 31 minutes when Craigan found the back of his own net following Nicolita's corner.
It should have been 2-0 two minutes later when Steaua were awarded a penalty after Ross Forbes fouled Romeo Surdu, however Ruddy saved Dorin Goian's power penalty.
Gannon made a change before half-time, as Steven Hammell took over from Ross Forbes, but the Steaua pressure continued.
Just as it looked as though Motherwell would survive at 1-0 down until the break, they conceded in the 45th minute.
Nicolita provided the low finish past Ruddy after Rafal Grzelak's pass.
Jim O'Brien came off the bench early in the second half, in place of Chris Humphrey, and the substitute created a chance almost instantly when his cross was headed wide by striker John Sutton.
Motherwell were looking to build on that effort, which represented progress, but 1986 European Cup winners Steaua had other ideas and went 3-0 clear in the 61st minute when Grzelak's cross from the left was headed in by Stancu.
Stancu should have had another goal but fluffed his shot, before Sutton again threatened at the opposite end.
This time his header demanded a save from Cristian Tatarusanu but from the rebound Coke blazed his shot over.
Ruddy, experiencing a busy debut, saved from Krum Bibishkov prior to Mihai Onicas hitting the post with a header.
The 22-year-old former Cambridge goalkeeper made another late save when he tipped Razvan Ochirosii's shot over the bar, on a miserable night for the Scots.