The Giallorossi had failed to beat their city rivals in their previous five meetings - including a 1-0 loss in the Coppa Italia final back in May - and this match looked to be heading the same way at the Stadio Olimpico until Federico Balzaretti gave Roma a 63rd-minute lead.
Lazio, playing as visitors, shot themselves on the foot when substitute Andre Dias was shown a straight red for a last-man challenge on Francesco Totti with nine minutes remaining.
Substitute Adam Ljajic then put the game beyond Lazio's reach when he earned an converted a penalty deep in stoppage time after he was brought down by Cristian Ledesma.
The result, Roma's fourth consecutive win in Serie A, lifted them to the top of the standings.
The loss was Lazio's second of the campaign.
Roma coach Rudi Garcia, experiencing his first derby, fielded a strong attack, with Totti in support of Gervinho and Alessandro Florenzi.
However, Roma struggled to break down their opponents and were limited to shooting from long range.
In the 12th minute, Daniele De Rossi tried his luck from distance and his attempt went over the bar.
On the half-hour mark, Totti's free-kick towards the far post found an unmarked Gervinho, who struck the ball way off target.
Before the break, Abdoulay Konko dribbled past two Roma players but lost control with Roma goalkeeper Morgan De Sanctis left to beat.
After a distinctly uninspiring first half, in which neither goalkeeper had a save to make, the game came to life after the re-start.
In the 51st minute, Roma were lucky not to concede an own goal when Michael Ciani's header from Hernanes' corner took a deflection off Roma defender Leandro Castan and hit the crossbar, with De Sanctis beaten.
Shortly after, Miroslav Klose wasted a great chance to put Lazio ahead when he got on the end of Senad Lulic's cross but fired wide from close range.
In the 62nd minute, Balzaretti sent a warning when his powerful shot from 30 yards struck the far post.
Seconds later Roma went in front when Balzaretti connected with Totti's corner and fired a left-footed volley from inside the box to beat Marchetti.
Dias' sending off hindered Lazio and it was Roma that doubled their lead before the final whistle.
The referee pointed to the spot following Ledesma's foul on Ljajic. The Roma man then sent Marchetti the wrong way as he struck the spot-kick towards the bottom right corner of goal to make it 2-0.