Arsene Wenger's side dominated for large periods, created many chances and should have added a couple more to Van Persie's solitary effort. Nevertheless, it was a vast improvement on Arsenal displays of recent weeks.
The key to their success on the night came in the form of the trio in attack - Bendtner, Samir Nasri and Van Persie.
None of the three occupied the conventional positions up front which confused Roma's backline from the whistle.
Bendtner largely played on the left but that did not stop him from bulldozing into the area whenever he saw the defensive door left ajar.
His forays almost paid big dividends on a handful of occasions.
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The lanky Dane's first effort suffered as a result of mis-control from Nasri's fine ball into the area, a good block inside the six-yard area then denied him a second bite before he smashed wildly over the bar from eight yards after the break.
One man who was more proficient in front of goal, when it mattered, was Van Persie.
The Dutchman's penalty strike came just before the interval, won when he was tripped by Mexes after being played into the area.
Van Persie's low, driven spot-kick gave Doni no hope in the Roma goal.
The second half opened in farcical fashion as Arsenal only had nine players on the pitch when they kicked off.
William Gallas was receiving treatment as the players were called out for the restart, but a superstitious Kolo Toure waited for his team-mate as he is always the last man to leave the dressing room.
The minor mishap did put the Gunners off their stride, and they had numerous chances to extend their lead in the second half.
Apart from Bendtner's wasted effort, Abou Diaby had a wonderful low strike tipped around the post by the agile Doni before Eboue flashed wide after pouncing on a mistake by Simone Loria.
Roma, who have only won one of their previous 15 games away to English sides, were a better proposition in the second half but provided little threat in attack, with Francesco Totti and Julio Baptista disappointing.
Wenger brought on Mexican left-sided forward Carlos Vela and Welsh youngster Aaron Ramsey in the hope that they could give his side a little bit more to play with at the Stadio Olimpico, but it remained at 1-0 with the promise of a nail-biting second leg.
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