Fazio made an awful mistake to gift Lazio an equaliser midway through the second half but, after Aleksandar Kolarov put Roma back in front with a free kick against his old club, made amends by heading their third goal.
It was Roma's seventh win in their last nine meetings with Lazio in all competitions and pulled them up to fifth place in Serie A with 11 points, one place and one point behind their opponents.
Lazio, winners of their last five matches in all competitions, dominated the early exchanges but Roma struck the first blow in the 45th minute.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
A long ball was launched into the area, Lazio defender Luiz Felipe and goalkeeper Thomas Strakosha collided and the ball ran loose to Lorenzo Pellegrini who cleverly backheeled it into the goal.
Pellegrini, who became the tenth different Roma player to score in Serie A this season, had been on the pitch for only nine minutes after replacing the injured Javier Pastore. It was all going smoothly for Roma until Fazio was dispossessed as he tried to dribble around Ciro Immobile and the Lazio forward ran on to fire the ball wide of Patrick Olsen for his first goal in a Rome derby.
But Roma were back in front within four minutes. Lazio substitute Milan Badejl tripped Pellegrini from behind on the edge of the area and Kolarov fired the free kick through a huge hole in the Lazio wall.
Argentina defender Fazio completed the win, and atoned for his earlier mistake, by meeting a free kick with a towering header to score four minutes from time.
(Reporting by Brian Homewood; editing by Clare Fallon)
