Roma, knocked out of the competition for the last two seasons by Manchester United and with a dismal record in England, had a few chances to pocket an away goal, but Arsenal should have built a bigger lead.
Nicklas Bendtner wasted their best chance shortly after halftime, blazing wastefully over the crossbar. Arsenal will hope Togo international striker Emmanuel Adebayor will be fit to return for what promises to be a tricky second leg.
"It's a regret," Arsenal manager Arsene Wenger told reporters when asked if his side should have killed off Roma. "I'm happy with the performance but there was room to score more goals tonight."
"The one positive, or not the only positive, is that we didn't concede a goal and it puts us into a position where we cannot just defend. We will want to attack and score goals over there."
After three consecutive 0-0 draws in the Premier League Arsenal began with real purpose at the Emirates and threatened to blow Roma away.
Without injured Cesc Fabregas in midfield, Samir Nasri took on the responsibility of pulling Arsenal's strings and his speed conjured an early chance for Bendtner.
Roma barely got out of their own half for 15 minutes but when they did Marco Motta forced a fingertip save from Arsenal keeper Manuel Almunia. Simone Loria then wasted a great chance for the Giallorossi as Roma began to settle.
As halftime approached Arsenal's dominance was rewarded when Bendtner played in van Persie on the left edge of the penalty area and Mexes, who was having a torrid night, got the wrong side of the Dutchman and committed a clumsy foul.
Van Persie confidently dispatched the penalty.
Arsenal began the second half in bizarre fashion as the game kicked off with centre backs William Gallas and Kolo Toure still walking down the tunnel, resulting in a yellow card for Toure.
On the restart the home side went for the kill, however, and had chances to extend their lead. Bendtner should have finished off a sweeping move and then Doni saved a low drive from Abou Diaby. Arsenal's profligacy in front of goal continued when Emmanuel Eboue shot wide after over-indulging.
The home side were almost made to pay when former Liverpool left back John Arne Riise shaved the post with a fierce cross shot but apart from a late flurry of pressure Roma were content to hang on.
"It was a fair result," Roma manager Luciano Spalletti said. "Their pace and intensity gave us a lot of difficulties but everything will be different in the second match."
