Levein's team get their bid to reach Brazil underway with a double header at the National Stadium against Serbia on Saturday before welcoming Macedonia on Tuesday.
Victory at home in the opening two Group A games would go a long way towards booking a place in the football extravaganza in South America in two years' time.
And Scotland legend McLeish, who won five out of six home matches while in charge of the national side, reckons home advantage is the key to success.
He said: "The occasions Scotland have managed to qualify for finals of major championships have been down to playing at Hampden in no small part.
"Looking back on my experiences as a player, I can remember the big games we won there against the likes of France and Spain, I can still recall the goals Mo Johnston and Kenny Dalglish scored.
"We just seemed to have a real confidence at Hampden with the Tartan Army there in numbers behind us.
"You often hear managers talking about the importance of fans to a team and it is not a cliche. The energy a player can get from fans can be tangible. It can give players an extra two or three per cent, in some individual's cases even more."
The ex-Rangers boss, who only lost to Italy at Hampden during his reign in charge back in 2007, reckons having two home games to kick off the Group A campaign is the perfect start for Scotland.
He agreed: "It is a good starter for 10. If Scotland can get off to a flyer with victories in the first two games it would be a marvellous boost.
"Everybody in the group will be hard to beat away from home so it is vital that we take as many points as we can at Hampden. Of course, you need the quality. You need to have a little bit about you. But Craig has some very good players."
McLeish, who will be at Hampden on Saturday working for Sky Sports, added: "When it comes to Scotland, the fans make a crucial difference.
"Visiting teams are intimidated by playing in front of a packed stadium filled with members of the Tartan Army."