The Spaniard vented his frustration at his sides’ inability to see the game out as their newly promoted opponents hit back twice to claim a 2-2 draw at the King Power Stadium
The Toffees twice took the lead through Aiden McGeady and Steven Naismith but were pegged back by strikes from Jose Ulloa and Woods.
And despite dominating possession, the former Wigan manager suggested that his side were guilty of resting on their slender lead.
“We should have won the game. We got in front twice and, if I had a criticism, it's that we should have gone more for a third goal with a little bit more brightness rather than just controlling the game," he said.
“I thought the tempo of the game dropped and, with four minutes to go, we didn’t show the concentration levels you need to keep a clean sheet. When you play a team like Leicester, you know they are going to come with a positive approach.
"It's difficult to speak about positives when you concede with six minutes to go when really we should see the game out. It's one of those that you feel is like a defeat. There's real disappointment that we didn't win.”
Martinez also confirmed that England international Ross Barkley has injured his right knee but could not confirm how long the midfielder would be out for.
"It could be more, it could be less than 8 weeks. We'll find out in the next few days. Ross has been through periods of adversity before and he will know how to cope with this injury.
"Ross is a phenomenal footballer and we are going to miss him. But the squad should be big enough to cope."
Meanwhile, Nigel Pearson was pleased with the efforts of his newly promoted Leicester side but refused to get carried away after a dramatic finale.
He added: “We’re disappointed to go behind twice but the players have shown a lot of character and ability to get ourselves back into it. It is always good to get something out of an opening game.
“We’ve shown resilience and belief and it’s good to get something out of a game against a side which has shown over a number of years the amount of quality that they have.We’re under no illusions as to how difficult this division can be. We will build on what has been a good point for us today.”