Max Tonetto was the man to suffer spot-kick misery for the Italians, as he ballooned his effort high over the crossbar to send the travelling fans wild.
The Gunners won the first leg 1-0 on home soil, thanks to a Robin van Persie penalty and the Dutchman led the attack again, alongside Denmark's Nicklas Bendtner.
Roma made a bright start to the game, although defender Juan suffered a hamstring injury which almost forced him off. However, he stayed on to open the scoring after just nine minutes.
Francesco Totti's mis-hit cross from the left was missed by both Kolo Toure and William Gallas and Juan was in the right place to fire home at the far post from six yards out.
The home side went close to doubling their lead midway through the first half, as Manuel Almunia first parried away a Marco Motta shot and then superbly tipped away a low piledriver from Rodrigo Taddei.
Arsenal's only effort on target was a weak Abou Diaby header from Gael Clichy's left-wing cross, as Arsene Wenger paced the touchline with worry etched on his face.
Right on half-time the home side thought they should have had a penalty when Motta went down under a challenge from Clichy, but referee Gonzalez waived the protests away.
The second half started slowly and it took until almost the hour mark before Emmanuel Eboue ran clear down the right, but slid his pass across the face of goal just too far in front of Bendtner.
Ten minutes from time former Gunners loanee Julio Baptista was left unmarked about ten yards out right in front of goal, but he slipped as he made contact and failed to send the ball anywhere near the goal.
Right at the death Arsenal almost snatched a dramatic clincher. Sub Theo Walcott lifted a ball to the far post, van Persie headed back across goal and Toure nodded over from close range.
Extra-time proved to be highly disappointing, with neither side threatening to make a crucial breakthrough as fatigue slowed the game to a crawl.
The shootout started with Eduardo having his effort saved by Doni, but Mirko Vucinic was then denied by Almunia as sudden death penalties were needed.
Eventually Tonetto was the man who cracked, as Arsenal made it four out of four English teams in the last eight of Europe's premier club competition.