Steve Clarke urged the Tartan Army to "anticipate success" and drive Scotland to the World Cup when they host Denmark in a winner-takes-all clash.
Scotland somehow emerged from a 3-2 defeat in Greece with the chance to seal top spot in Group C on Tuesday.
Leaders Denmark lost a lead against bottom side Belarus inside three second-half minutes before sealing a 2-2 draw.
The Danes had beaten Belarus 6-0 on neutral territory last month and few had countenanced the possibility they would slip up in Copenhagen.
Scotland needed a draw in Piraeus to guarantee themselves the chance to leapfrog Denmark in the final game and very nearly got one despite going three goals down.
Goals from Ben Gannon-Doak and Ryan Christie set up a thrilling finale with Scott McTominay denied by an outstanding stop, before an agonising five-minute wait for the result in the Parken Stadium to be confirmed.
Clarke now wants his players to maintain their momentum and fans to take advantage of the reprieve and deliver the Hampden Roar.
"We spoke about that in the dressing room," he said. "Although we lost the game, we're probably finishing in a good moment with good momentum in the game.
"If we can carry that into Tuesday night with the help of a full Hampden and hopefully a very positive Hampden right from the first kick-off...We have a great group of players, but we're going to need backing from the first minute at Hampden."
Scotland did not always get a positive atmosphere in their other two home games. They were booed by some fans at half-time in the visit of Greece and were outplayed for about an hour before triumphing 3-1.
Some even jeered after a 2-1 victory over Belarus.
"We need positive backing," Clarke said. "One of my favourite sayings for the players is 'play with the anticipation of success and not the fear of failure'. I'm going to ask the Hampden crowd to do that on Tuesday night.
"We need them for the first minute to be with us and in the difficult moments in the game to be especially with us. That's what this group of players deserve and, if they get that, I'm pretty sure we can do something special.
"I think we can inspire each other. The team can inspire the fans and the fans can inspire the team."
Having struggled in periods of their last three matches, Scotland still find themselves with the chance of qualification for the first time since 1998.
And with the guarantee of a play-off to fall back on, the potential rewards are greater than the risk of failure.
Clarke, who is looking to add to two European Championship qualifications, said: "I feel as though I've been in charge of quite a lot of big games and that's a credit to the players because they've put us in a lot of positions for big games.
"We've met a lot in terms of qualification, tournament games, lots of big, big games.
"This group of players have given their country so many big games to look forward to, so many big nights. Tuesday night will be another big night and hopefully a big result for us."