The Spaniard watched Everton start positively at the King Power Stadium on Saturday and deservedly take the lead in the 20th minute thanks to Aiden McGeady's stunning effort.
However, Leonardo Ulloa levelled for the Premier League new boys two minutes later as Leicester fought back on the opening day of the new season.
Steven Naismith put the visitors back in front before the break and that looked to be enough until Chris Wood's 86th-minute strike earned a share of the points for Leicester.
Martinez, who lost star midfielder Ross Barkley to a knee ligament injury 24 hours before kick-off, was pleased with Everton's display, but was disappointed they allowed Leicester back into the game on two occasions.
"We should have won the game, and we only have ourselves to blame really," he said. "In the first half we did everything we had to do.
"We controlled the environment, you could expect a newly promoted side with the fans behind them to have a bright approach and we coped with that really well.
"We did the hardest thing which is get yourself in front twice, [but] the two goals we conceded, the first one is unfortunate, it is just a rebound from a corner and the second one we got too relaxed.
"If I've got a criticism, it's that we should have gone for that third goal a little bit more, rather than just controlling the game.
"I thought the tempo dropped a lot, with six minutes to go we didn't show enough concentration."
Barkley has undergone a scan on his injury and the results have proved inconclusive.
"We don't know how bad the injury is," added Martinez.
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