Skriniar, one of the players at fault for the Verona goal, made amends by heading Inter back in front in the 69th minute.
"This is the seventh consecutive win and that must give us conviction and make us understand that we must continue on this path," said Inter coach Antonio Conte. "Today, we showed a great will to win."
"This is certainly a more balanced championship than in previous years. It will be very difficult, all the teams have strengthened. You can lose points against anyone, you must always be on the ball."
Inter produced a lethargic first half but came to life after halftime and Martinez got in front of his marker to volley in from Achraf Hakimi's cross.
Verona, surprise contenders for a European place this season, levelled after goalkeeper Samir Handanovic mishandled Davide Faraoni's cross, Skriniar failed to react in time and Ilic poked the ball home.
It was all forgotten six minutes later, however, when Skriniar scored with a looping header, only his fifth goal in three-and-a-half seasons at Inter.