It was Ange Postecoglou side's fourth successive draw and left the Socceroos third in Group B on 10 points after wins for Saudi Arabia and Japan put them three points clear in the top two automatic qualifying places.
"We've had a tough road up until now," Leckie said. "We have four games now with three at home, so this is a time for us to step up.
"We dominate most of our games at home. We need a lot of character amongst the team and a lot of players to step up and if we can do that I'm very confident we'll get all our results at home."
Australia host the fourth-placed United Arab Emirates in Sydney on Tuesday. Their only remaining game away from home is against Japan on Aug. 31.
Leckie's header in Tehran was a welcome score from open play after Australia's previous three goals in the 1-1 draw with Japan and 2-2 draw with Thailand had come from Mile Jedinak penalties.
The target man said Aaron Mooy's corner had been executed perfectly.
"There were a couple of boys in that area that dragged some boys away and I was able to get pretty free," Leckie added.
"It came off the head perfect and ended up in the top left so I'm very happy with that on a personal level but ultimately taking the three points would have been the best reward."
Both Leckie and debutant midfielder Jackson Irvine felt that if the team had scored a second goal, the game would have been killed off before Ahmed Yasin grabbed Iraq's equaliser from close range with 14 minutes remaining.
"If we'd carried on and got the second before half-time or straight after half-time, it might have killed the game a little bit more," Jackson said.
"We offered them a chance to come back into the game and a team with nothing to lose can throw everything at you and that's what they did and that led to them getting the equaliser.
"We're disappointed overall, but there are a few good signs to take from the game."
(Writing by Greg Stutchbury in Wellington; Editing by Nick Mulvenney)
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