Belgium striker Lukaku guided a header into the top corner in the first half before his Argentine partner finished off a counter-attack to continue the Nerazzurri's charge towards the title.
Sassuolo dominated possession at San Siro but were frustrated by a stubborn home defence until Hamed Traore curled home a superb finish five minutes from time.
"It was very important for us, to climb up the table and increase the gap," Martinez said after the match. "At the start of the second half of the season we had a talk, and now we're doing everything that the coach wants."
With nine matches left, Inter are firmly on course to win their first Serie A title since 2010 with 71 points, 11 ahead of second-placed AC Milan. Sassuolo remained ninth with 40 points.
The fixture was postponed on March 20 after a COVID-19 outbreak in the Inter squad.
The rescheduled date gave Antonio Conte's side a chance to stretch their lead, and they wasted little time as Lukaku's glancing header found the top corner from Ashley Young's cross after 10 minutes.
Jeremie Boga was denied by a last-ditch Milan Skriniar block and Pedro Obiang's shot was held at the second time of asking by Samir Handanovic, but the visitors could not make their 68% possession count.
The hosts hit on the counter for their second after 67 minutes, as Lukaku found Martinez with a pass and the striker arrowed a shot home.
Sassuolo found a way back into the match in style when Traore struck a half volley into the top corner.
Inter went close to a late third as Sassuolo pushed for an equaliser, substitute Alexis Sanchez's lob landing on the roof of the net and Lukaku having an effort disallowed for offside.
Conte led the Juventus to the first three titles of their nine-year run, before taking over the Inter job in 2019 with the aim of shifting the balance of power back to Milan.
"To think that in two years we have made up so many points on the dominant force of the league for the last nine years, I could not have imagined it in my wildest dreams," Conte said.
"It does not mean we are better than Juve, who have a very strong team. We rolled up our sleeves and started to work in an important way towards an objective, to be credible.
"The league was always dominated by one team and we have managed to change that. That has motivated us a lot, it was a boost for everyone, especially for the many guys who are fighting for this goal for the first time.
"Last year we played in a Europa League final, now we are aiming to achieve the maximum objective."