Liverpool fell behind to Bruno Fernandes’ header in the 19th minute and for a while they looked rattled.
However, a Javier Mascherano screamer and close-range effort from Ryan Babel turned things around before the interval, with Steven Gerrard’s 33rd European goal wrapping things up after the break.
It was the first time since the 6-1 win over Hull in September that Liverpool had scored more than twice in a game, a run of 29 matches.
A year ago to the day they beat Real Madrid 1-0 in the Bernabeu on the way to a 5-0 aggregate win before their eventual Champions League quarter-final defeat by Chelsea.
But their early struggles in the Steaua Stadium showed just how far the club’s fortunes have plummeted in 12 months as Unirea created a number of chances, mainly from free-kicks and corners, which the visitors regularly failed to deal with.
The first example of that came in only the second minute when Jamie Carragher’s header from Iulian Apostol’s free-kick only reached the edge of the penalty area and Sorin Frunza lashed a dipping volley onto the top of Jose Reina’s net.
Less than 60 seconds later, Lucas Leiva picked out Gerrard with a through-ball but the midfielder’s fierce low drive was parried into the air by goalkeeper Giedrius Arlauskis.
Liverpool were retaining possession for long periods without really hurting their opponents and it was to cost them as, in the 19th minute, Unirea levelled the tie on aggregate.
Daniel Agger’s last-ditch tackle on Marius Onofras gave the hosts a corner from the left and Razvan Paduretu’s inviting delivery was perfect for centre-back Fernandes to power home a header virtually unchallenged.
Nine minutes after, George Galamaz’s foul on Martin Skrtel resulted in the Unirea captain coming off worst and, despite trying to run off his injury, he was replaced by Ersin Mehmedovic.
Then on the half-hour mark Liverpool seized control of the tie from the most unlikely of sources.
Yossi Benayoun and David Ngog combined to allow Carragher to cross and, although Babel mis-controlled Gerrard’s far post header with only the goalkeeper to the beat, the ball broke to Mascherano outside the penalty area to brilliantly lash a shot past Arlauskis.
However, the Reds almost handed the initiative straight back to Unirea in the 36th minute when Paduretu’s inswinging corner caused chaos in the six-yard area, hitting Agger but bouncing inches wide.
Paduretu then had Reina diving to his right to tip around the post with a 25-yard effort but Liverpool gave themselves the cushion they desperately needed five minutes before half-time when Gerrard swung in a free-kick from the right and Babel took one touch before swivelling to fire home from close range.
Liverpool picked up at the start of the second half and Gerrard fired wide of Arlauskis’ right-hand post before forcing him into a low save on the same side soon after.
And if there was any doubt about Unirea staging an unlikely comeback, it was snuffed out by Gerrard in the 57th minute.
Babel broke down the left, moving the ball to Lucas and he picked out Benayoun, whose mazy dribble into the area was only half-stopped, allowing Gerrard to run onto the loose ball and blast a low shot under Arlauskis.
That strike was his 33rd in Europe, taking him past Alan Shearer, who scored 30 for Newcastle and two for Blackburn.
That prompted the introduction of youngster Martin Kelly for Carragher at right-back and, almost immediately, Sotirios Kyrgiakos after Unirea substitute Antonio Semedo caught Skrtel in a challenge.
With thoughts turning to Sunda’s Premier League clash with Blackburn, Rafael Benitez also replaced Benayoun with Fabio Aurelio for the final 15 minutes.
Emiliano Insua’s late intervention poked the ball off the toe of Paduretu just as he was about to pull the trigger in the six-yard area but otherwise there were no other scares.
But their first-half defensive lapses will not have gone unnoticed by Rovers boss Sam Allardyce ahead of his side’s visit to Anfield.
Much greater challenges lie ahead for Benitez and his Liverpool team.