Defeat to Juventus before the international break shunted Antonio Conte's men down to second in Serie A but a superb performances from the front two trimmed the margin at the top back to a point.
Martinez made an early incision and Lukaku struck twice in seven minutes to establish a two-goal advantage at half-time, Domenico Berardi having fleetingly restored parity for the host.
The exceptional Martinez won the penalty for his strike partner's second and, in the 71st minute, dispatched a spot-kick of his own, although the visitor was made to fight for victory as substitutes Filip Djuricic and Jeremie Boga set up a grandstand finale.
Martinez's unerring finish from the edge of area in the second minute set the tone for a frenetic first half.
The Argentinian striker skewed a second chance wide and Sassuolo made the reprieve count through winger Berardi, who angled an excellent drive across Samir Handanovic in the 16th minute.
Martinez did soon have the ball in the net again, only for the goal to be chalked off due to Lukaku's foul on Federico Peluso.
A second disallowed strike followed, Francesco Caputo straying offside prior to a tidy finish, at which point Lukaku took command.
The striker cushioned Stefan de Vrij's pass, turned and fired Inter back ahead before converting from the spot in the 45th minute after Marlon dragged Martinez to the turf.
The strength of understanding between Inter's attackers was no clearer than when Lukaku, on a hat-trick, granted his team-mate the opportunity to dispatch a second.
Djuricic dived in on Nicolo Barella and Martinez converted the resultant spot-kick, although the Sassuolo midfielder went some way to making amends by poking in his side's second three minutes after the indiscretion.
Boga then weaved away from a couple of challenges and coolly found the corner nine minutes from time, but the Nerazzurri did just enough to get back to winning ways.