However, Germany were back in front in the 70th minute when Muller fired in his second from close range, and there was more disappointment for Scotland as Charlie Mulgrew received a second booking just a few moments before the final whistle.
Below, Sports Mole takes a closer look at the Group D meeting.
On target: 9
On target: 3
Was the result fair?
On balance, Germany were the better side in Dortmund. Scotland could have quite easily picked up a point on another day, but there was no doubt that Germany had dominated proceedings in front of their supporters. If they turned their domination into clear chances, the gap between the two sides would have been much wider.
Joachim Low will definitely feel that his side could play a lot better than they did in this match. In truth, the world champions rarely got out of third gear, and showed only a few glimpses of finding their top form. With so many key players out of action, it was a challenge for Germany, but the likes of Muller, Toni Kroos and Marco Reus all played crucial roles.
Very promising. Gordon Strachan has worked wonders since becoming manager, and Scotland took another step forward in a losing cause tonight. They'll be bitterly disappointed to walk away without a point, but they impressed while being on the back foot for long periods. The visitors defended admirably as a unit, and were always a threat when breaking in numbers.
Sports Mole's man of the match
Thomas Muller: It wasn't really a night for individuals in Dortmund, but there was no doubt who made the biggest contribution. Muller was always a threat when he collected the ball in areas further up the pitch, and he made the difference with two goals. It could have been a hat-trick for the Bayern Munich forward, but he was denied by the pitch late on.
Charlie Mulgrew will be suspended when Scotland return to action next month, and he managed to find the most foolish way to be sent off. With Scotland trailing in stoppage time - and after being booked just moments earlier - Mulgrew reacted to a German free kick by kicking the ball away in frustration. A second yellow card followed, and he was forced off the pitch.
Svein Oddvar Moen was forced to dish out five yellow cards and a red in a fiercely competitive encounter, but it was difficult to argue with many of his decisions until he blew the final whistle as Scotland were preparing to take a corner.
Germany: Low's side are back in qualifying action against Poland on October 11.
Scotland: October 11 will also see Scotland continue their qualifying campaign when they host Georgia.