Lukaku's Sunday double raised his tally to seven goals in eight matches across all competitions since joining the Giallorossi in August on a season-long loan.
"A lot of things were said about me but I worked hard over the summer," Lukaku, previously accused of no longer being at his physical peak, said.
It was not all good news for Jose Mourinho's side, though, with fellow front-man Paulo Dybala - who missed long stretches of last season through injury - exiting the match in the first half with a left knee problem.
Lukaku used his chest to redirect in a cross from Rick Karsdorp following a set up from Leonardo Spinazzola 20 minutes in, doubling Roma's lead after Houssem Aouar had put the visitors ahead 60 seconds earlier.
The 30-year-old Belgium international added his second three minutes from time.
Andrea Belotti also scored for the Giallorossi after replacing Dybala, while Nahitan Nandez converted a late penalty for Claudio Ranieri's Cagliari.
Lukaku said Roma coach Mourinho was fundamental in his decision to move to Italy's capital.
"I have a special relationship with (Mourinho). He knows my family. He knows my kids," Lukaku said.
"I really have confidence in him and he really has confidence in me."
Defending champions Napoli were beaten at home for the second time in the league this season, falling 3-1 to Fiorentina.
Fresh from his first call-up by Italy in three years, Giacomo Bonaventura scored the Viola's second goal midway through the second half.
The 34-year-old Bonaventura has been involved in the most goals among Serie A midfielders this season, scoring four times and assisting twice.
Josip Brekalo had put Fiorentina ahead early on, before Victor Osimhen equalised with a penalty in first-half stoppage time.
Substitute Nico Gonzalez sealed it for Fiorentina late in the second half.
Fiorentina moved up to fourth, three points ahead of fifth-placed Napoli.
Lazio threw away a two-goal lead but still managed to beat visiting Atalanta 3-2 at the Stadio Olimpico.
An own-goal from Charles De Ketelaere and a strike from Valentin Castellanos put the hosts 2-0 up after just 11 minutes.
Ederson and Sead Kolasinac drew Atalanta level before subsititute Matias Vecino won it for Lazio seven minutes from time with an acrobatic half volley.
Monza moved up to seventh with a 3-0 win over struggling Salernitana, with goals from Andrea Colpani, Samuele Vignato and Matteo Pessina.
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