Emmanuel Adebayor scored the only goal of the contest in the 65th minute, showing brilliant skill and strength to fire low after a quickly-taken free-kick from Kyle Walker, but there could have been a chance of a late equaliser.
Seamus Coleman appeared to be felled in the area by Etienne Capoue late on but referee Mark Clattenburg waved away the protests as the Toffees were left to rue what might have been.
"I haven't seen the replay but it looked a very sloppy incident," Martinez said after the match.
"I think it was a careless challenge. He tries to win the ball and didn't at all.
"I'm not sure if there was enough contact for a penalty, but I've seen them given.
"You need a little bit of luck and it was one of those days for us."
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