Luciano Spalletti's side claimed a 2-0 victory at the Stadio Olimpico thanks to second-half goals from Kevin Strootman and Radja Nainggolan.
Tempers flared in a typically feisty meeting of the capital clubs, with six players ending up with yellow cards and unused substitute Danilo Cataldi earning a straight red for his part in a fracas that took place in the dugout after Strootman's goal.
But Lulic's post-match comments about what he considered to be taunts from Rudiger have prompted a backlash, amid accusations of racism.
He told Mediaset Premium: "Rudiger was provoking us even before the game. Two years ago, he was selling socks and belts in Stuttgart. Now he acts like he's some phenomenon.
"It's not his fault, really, but the fault of those who are around him. They haven't taught him good manners."
When asked to evaluate his remark and possibly apologise, Lulic added: "I said what I think."
Lazio were quick to apologise for Lulic's comments, admitting that the Bosnia international had gone "over the line".
"The club is pained by these comments and the way they were interpreted," said Lazio director of communications Arturo Diaconale. "I apologise on behalf of the club, these are statements made in the heat of the moment by a player who just lost the derby.
"It went over the line. The controversy started with previous comments from the Roma player, but we all need common sense and to not blow this up."