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The in-form Martinez, handed the captain's armband by Simone Inzaghi, took his goal tally for the season to 15 by opening the scoring with a header in a totally one-sided first half at San Siro on Monday (AEDT).
Milan failed to register a shot on target in a lacklustre display that increased its winless run to seven matches, putting more pressure on head coach Stefano Pioli.
Second-placed Inter moved five points clear of its city rival behind runaway Serie A leader Napoli, with the Rossoneri down in sixth spot as their dismal start to the year continued.
Martinez forced a magnificent reflex save from Ciprian Tatarusanu and also headed just wide in a blistering start from the Nerazzurri.
Inzaghi's side was dominant, but referee Davide Massa correctly refused to award it a penalty when Edin Dzeko went down too easily under a challenge from Matteo Gabbia.
It seemed to be only a matter of time before Inter were in front, though, and the opening goal duly arrived in the 34th minute, when the lively Martinez rose above Simon Kjaer to nod in Hakan Calhanoglu's corner from the left.
It remained one-way traffic following the break, Tatarusanu palming away Martinez's shot at his near post after Pioli introduced Rafael Leao and Alexis Saelemaekers.
Olivier Giroud finally had Milan's first attempt after 57 minutes, but his header sailed over the crossbar and was never going to trouble Andre Onana.
Further goals instead appeared more likely at the other end.
Romelu Lukaku had one ruled out when he was deemed to have dragged down Malick Thiaw, and Martinez had another chalked off for a marginal offside, but his first-half finish proved decisive.
