Inter had failed to win any of their four most recent pre-season friendlies but they collected three ICC points on a poor pitch in Lecce to move off the bottom of the competition's table.
Luciano Spalletti's side struggled throughout the first half of Saturday's clash and Samir Handanovic made three good saves to allow his team to go into the interval level.
However, seven minutes after the restart an improved Inter took the lead, Martinez providing a sweet first-time finish after an excellent Dalbert raid down the left flank.
Lyon, who had Memphis Depay, Bertrand Traore and Houssem Aouar on the bench, rarely showed any of their first-half pep and verve following the interval, with Handanovic mostly untroubled as he collected a well-earned clean sheet.
Rafael's curling strike forced a brilliant fingertip save from Handanovic as Lyon carved open the Inter defence in the right channel thanks to a smart backheel from Jordan Ferri.
Lyon were creating the better chances and Handanovic had to make a smart stop to his left in the 25th minute, the Inter goalkeeper pushing away a strong header from Martin Terrier.
Handanovic was called into action again before the break, denying Mariano Diaz after the former Real Madrid striker turned Milan Skriniar far too easily to run on to a useful Ferri pass.
Inter finally mustered a shot on target in first-half stoppage-time, Mathieu Gorgelin comfortably dealing with a speculative 25-yard shot from Yann Karamoh.
The Serie A side kicked on following the interval. Matteo Politano's cross found Martinez, but the new signing was only able to direct his header into the ground and over the crossbar.
But the striker found the net with his next chance to open the scoring in the 52nd minute.
Dalbert's low left-wing cross was cleverly dummied in the box by Politano, allowing Martinez to sweep home at the back post.
The Brazilian left-back perhaps should have repeated the trick but instead tried to beat Gorgelin at his near post and could only fire into the side netting from a tight angle.
Inter's defensive shortcomings may have been displayed regularly before the break, but they were not shown again throughout the second half with Bruno Genesio's team unable to create a clear opportunity to equalise.
Substitute teenage striker Amine Gouiri sent an acrobatic strike flying over the crossbar but Inter held on for victory as Spalletti's preparations for the new Serie A season continue.