Blues found themselves two goals behind just 23 minutes into the match against David Moyes' side but clawed their way back with goals from Cameron Jerome and Craig Gardner to gain a point from the game at St Andrews on Saturday afternoon.
"I reminded the players before the game how well they've done and they should relax a little bit but not be complacent but I think they went into a kind of deep sleep for the first 20- 25 minutes of the game and Everton raced into the lead," the Blues boss told Sky Sports after the match.
"But we fought back brilliantly again and we showed the character that I think has been indicative of this team the whole season."
"We got the goal (Jerome's) back and it rocked Everton a little bit, they'd dominated early proceedings, had chances, Joe Hart had a great save, that was the early warning sign which we didn't heed.
"We were just slack, we gave Everton too much respect but they are a quality side and for us to fight back and get a point against a team like Everton I'm so proud of the guys."
The result meant that Blues remained eighth in the Premier League table and McLeish refused to rule out the possibility of a seventh place finish and qualification for Europe.
"You have got to believe anything is possible," he said.
"Can we stay above Everton? We will do our best, we'll try, we'll definitely try.
"You see that team and the effort they have made this season, they won't rest and they'll keep trying to get points but we have tough games coming up, quality games, games that we are relishing, games why we love being in the Premier League."