Martinez's men took the lead at Goodison Park thanks to a 57th-minute strike from Steven Naismith before two goalkeeping errors from Tim Howard gifted Leicester the lead with goals from David Nugent and Esteban Cambiasso.
Matthew Upson deflected Romelu Lukaku's near-post header into his own net in the 89th minute to ensure a fortunate point for Everton.
Despite that late leveller, Martinez is keen for Everton - who beat Young Boys 4-1 away in the UEFA Europa League on Thursday - to adapt better to playing shortly after returning from continental trips.
"Drawing games is frustrating but as a team we gave everything today," Martinez said in quotes reported by BBC Sport.
"It's been a demanding week and we need to learn about coming back from Europe, but that is not going to happen overnight.
"When you come back from Europe - you have seen with other teams around - it can be very difficult."
Everton have won just one of their last nine Premier League games and remain only six points above the drop zone. And, although Howard was directly to blame for both goals, Martinez wants his entire defence to improve.
"We need to defend well in the box because at two moments in the match we conceded soft goals," he added.
"Everyone is in the scrap until you get 40 points, that's the way the league works. Our focus is to get as high up the league as we can."
