United played the last quarter of the game with 10 men after defender Nemanja Vidic, who returned to the starting line-up for the first time since suffering a calf injury in August, was sent off for a foul on Gabriel Giurgiu.
Rooney stayed calm to score from the spot in the 64th minute and he repeated the feat in stoppage-time as United laboured to a much-needed victory after draws in their first two group matches.
The home side also finished the game with 10 men after Milan Perendija was sent off in the dying minutes.
United, runners-up to Barcelona in last season's competition, climbed to second in the standings with five points from three games. Leaders Benfica have seven points.
"I'm satisfied with the result and the performance," United manager Sir Alex Ferguson told Sky Sports. "We lacked penetration in the first half but we stayed patient and got the result we needed. It's not easy playing away from home."
Ferguson started with Rooney and Javier Hernandez in attack after resting the pair in Saturday's 1-1 Premier League draw at Liverpool but they struggled to create chances as the Romanian side defended strongly.
Michael Carrick blazed United's best chance over the bar in the first half and Otelul had opportunities to take a shock lead, the lively Ioan Filip's fine solo run ending with a rasping 20-metre shot over the bar.
Rooney missed a good chance, allowing keeper Branko Grahovac to easily smother his feeble free-kick from the edge of the penalty area and Patrice Evra also failing to trouble the Bosnian keeper with a low shot.
The Romanian champions responded in style when Liviu Antal headed just over the bar from Sorin Frunza's inswinging corner.
There was a surprising lack of urgency though from the visitors after the interval with Otelul's full-backs Adrian Salageanu and Cornel Rapa comfortably containing wingers Antonio Valencia and Nani.
But United continued to dominate possession and the breakthrough finally came when Otelul defender Sergiu Costin needlessly handled Rooney's cross and the England striker stepped up to convert the resulting spot-kick.
Vidic was sent off minutes later and Otelul sensed a chance to get back in the match but United kept hold of the ball well and Otelul were also reduced to 10 men when Perendija was sent off after receiving his second yellow card for fouling Hernandez.
Antal then tripped Rooney in the area in stoppage-time and the United striker steered his second penalty into the corner of the net to secure the points.
Ferguson said United would probably not appeal against Vidic's red card.
"He's raised his foot and in the context of the referee's interpretation when you play in Europe, I can see why he has given him the red card, but it is very harsh I think," he said.
"I don't think you win these things."
Otelul coach Dorinel Munteanu was heartened by his team's display.
"It was an important experience for my team and I think we performed well," said
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