Shaky defending from set-pieces in the first half saw them fall behind to Bruno Fernandes’ header in the 19th minute and for a while they looked rattled.
However, goals from Javier Mascherano from long range and Ryan Babel from close in turned things around before the interval and Steven Gerrard’s 33rd European goal after the break ensured any potential embarrassment was avoided.
“We showed character. It was a difficult game because of the pitch. We needed to be calm and do the job properly,” Benitez said. “To play better we needed to score our goal as they were dangerous at set-pieces.
“I think once we scored a goal everything was easier and the second and third goals made things better.’’
However, just as he thought his injury worries were easing, Benitez lost centre-back Martin Skrtel to a suspected broken foot after a second-half clash with substitute Antonio Semedo.
“At this moment we don’t know how serious it is but it could be a broken foot,” said the Spaniard.
“That would be bad news but we will have to wait and see what happens but it was a bad tackle.”
Gerrard’s goal took him past Alan Shearer (with 30 goals for Newcastle and two for Blackburn) as Britain’s leading scorer in Europe.
Benitez felt it was an honour well deserved. “It is really positive for him and the club – it also means we are playing a lot of games in Europe so hopefully we can play a lot more and he can score more goals,” said the Spaniard.
“It is very good news for the club. This is a historic club with a lot of records so to have a new record is massive for him and very important to the club.’’
Unirea coach Roni Levy admitted even when his side opened the scoring he did not consider there could be a potential upset.
“I knew it was an early goal and we had a long time until the end of the game,” he said.
“The equaliser came very early so I did not have much time to think about what our goal meant.
“From my opinion we deserved a better result.”
UNIREA URZICENI: Arlauskis, Galamaz (Mehmedovic, 27), Fernandes, Bordeanu, Maftei, Paraschiv (Gomes, 56), Paduretu, Frunza, Apostol, Bilasco, Onofras (Semedo, 62).
LIVERPOOL: Reina, Agger, Insua, Carragher (Kelly, 62), Skrtel (Kyrgiakos, 66), Gerrard, Benayoun (Aurelio, 78),Babel,Mascherano, Lucas, N’Gog.
Referee: Stefan Johannesson.