John Souttar, who has battled back from two career-threating achilles injuries, scored a fairytale opener, described by Clarke as “the best story ever”, before a Che Adams finish five minutes from time secured the victory that put the national side among the six best-placed group runners-up for qualification for Qatar 2022, ending Denmark's perfect group record in the process.
The 23-point tally equals Scotland' s best ever record in qualification and means the team will have home advantage against an unseeded nation in their play-off semi-final.
In a double boost, the five Scots who started the match on a yellow card all managed to avoid picking up the caution that would have ruled them out of the opening play-off fixture.
It means, barring injury, captain Andy Robertson, Stephen O'Donnell, John McGinn, Billy Gilmour and Che Adams will all be available when the big match comes around on either March 24 or 25.
Securing seeding status ensures Scotland will avoid a potential semi-final fixture against the likes of Cristiano Ronaldo's Portugal and Euro 2020 winners Italy, both of whom dropped into the play-offs after failing to win their respective groups.
Grinning from ear-to-ear, Clarke told Sky Sports: "I’ll just start with a smile. I always get criticised for not smiling but if you can’t smile after a performance like that from your players then there’s something wrong with you.
“I’m delighted. Everything came together and it was a good performance.
"We played good football, always on the front foot trying to win the game. It was a good night for us.
"It was a great night, another great atmosphere at Hampden. Fantastic crowd got behind the team when we started to flag a little bit second half, which is really important. It gave the boys a little boost and we got the second goal to make it more comfortable.”
Clarke hailed the contribution of Hearts defender Souttar, adding: “Best story ever. I’m so happy for him.
“He’s had a horrendous time with injuries to come back and even be in the squad.
“I know he was delighted to just be back with us but to get the opportunity to play, and to play the way he did, not just the goal, defensively and with the ball, it’s a great story.”