Robert Snodgrass enjoyed a perfect debut start for Scotland as he headed home the winner against Denmark.
A deflected Charlie Adam shot put the Scots ahead after 21 minutes, before an Allan McGregor error allowed Christian Eriksen to equalise.
But Snodgrass restored the advantage just before half-time with a superb headed goal from six yards.
The victory is Scotland's first over the Danes in six matches, ending a winless streak going back to 1975.
The visitors, ranked 21st in the world, enjoyed the lion's share of possession in the early stages, with Scotland initially restricted to long balls to lone striker Kenny Miller.
Scott Brown, yet to play a competitive match for club side Celtic this season, had to be substituted 20 minutes into the game.
The Celtic skipper was involved in a collision with Simon Kjaer minutes earlier, with manager Craig Levein choosing to remove the former Hibenian man from the match rather than risk an injury.
The deadlock was broken three minutes later, with Scotland taking the lead against the run of play.
Adam fired a free-kick in from the left-hand side of the Denmark penalty area and the ball was helped into the net from a deflection off William Kvist Jorgensen - thus robbing the Liverpool man of what would have been his first international goal.
The slender lead disappeared in embarrassing circumstances nine minutes later.
Scotland's first-choice keeper McGregor came off his goal-line to punch away Eriksen's free-kick from wide on the left, only to see it drift over his head and into the net.
The lead was restored two minutes from the interval, with a superb passing movement linking Gary Caldwell, substitute Don Cowie and lone striker Miller.
The Cardiff striker then picked out Snodgrass in the Denmark penalty area and he stooped low to head Scotland back ahead.
Torrential conditions continued into the second half, with Denmark looking more comfortable in their attempts to control the ball on a greasy surface.
In the 53rd minute McGregor did well to tip a long-distance drive from Denmark's half-time substitute Lasse Schone over the bar for a corner.
And with the Danes in almost total control of the second half, Levein rang the changes: withdrawing Miller and Adam and introducing Graham Dorrans and Craig Mackail-Smith.
And Mackail-Smith almost made an early impact. Niki Zimling's backpass became trapped by puddles of rainwater inside the Danish penalty area, but with the Brighton striker ready to pounce, goalkeeper Thomas Sorensen was able to block the shot.
Levein introduced James Forrest of Celtic, Barry Bannan of Aston Villa and Blackburn's Grant Handley in the closing stages, as terrible weather on the south side of Glasgow removed much of the sting from the fixture.
The Danes persevered right up until the final whistle - Sorensen came up for a corner in the last seconds of the match - but Scotland held out for a first win over the Danes in 36 years.
Scotland: McGregor, Bardsley, Caldwell, Wilson, Crainey, Morrison (Bannan 67), Brown (Cowie 21), Adam (Dorrans 57), Snodgrass (Hanley 88), Naismith (Forrest 74), Miller (Mackail-Smith 57).Subs Not Used: Bell, Berra, Robson, Goodwillie, Turner, McNaughton.
Goals: Kvist Jorgensen 23 og, Snodgrass 44.
Denmark: Sorensen, Christian Poulsen (Zimling 46), Agger (Jorgensen 58), Boilesen, Kjaer, Kvist Jorgensen, Eriksen, Jacobsen (Silberbauer 73), Rommedahl (Schone 46), Krohn-Delhi (Kadrii 76), Bendtner (Pedersen 46). Subs Not Used: Lindegaard, Kristensen, Mads Junker, Mads Junker, Enevoldsen.
Goals: Eriksen 31.
Ref: Marco Borg (Malta).