The odds were stacked against Scotland as Gordon Strachan’s men looked for their first win on German soil since 1999.
Backed by a vociferous Tartan Army, they found themselves under the cosh within nine minutes of the referee’s opening whistle as an unmarked Muller headed wide from six yards out.
Scotland were holding firm, though, and looked as though they may cause the world champions problems on the counter-attack, with Barry Bannan seeing a strike deflected behind after being found in the area by James Morrison.
The hosts then immediately headed up the opposite end and Erik Durm saw a long-range effort palmed behind by David Marshall, but the home crowd sensed an opener was on its way.
That breakthrough came on the 18th minute and the lively Muller was the man on target as he rose to meet Sebastian Rudy’s cross from the right and steer a looping header beyond the helpless Marshall.
Marshall was then called into action as he pushed away a bending effort from Reus before Russell Martin smothered Andre Schurrle on the rebound as Strachan’s side were forced to dig deep.
Scotland were being limited to counter attacks and Germany continued to press in search of a second as Marshall was there again to claw away Schurrle’s cross, which Grant Hanley had deflected towards the back post.
The visitors’ best chance did come on the break as Ikechi Anya was found by Steven Naismith, who clipped a ball over the top for the Watford man to run onto. He muscled away from the last defender and squeezed a strike away, but Manuel Neuer was equal to his soft half-volley.
Scotland came out from the break full of intent and a surging run forward from Alan Hutton was inches away from resulting in an early equaliser.
The Aston Villa full-back picked out Naismith, who twisted and turned away from Jerome Boateng and Benedikt Howedes, before clipping the back post with a low effort.
Germany were keen to get back on the front foot and Schurrle was left frustrated as Marco Reus’ wicked free-kick was headed clear into his path, but he could only lash his volley well over the bar.
Scotland appeared to be playing with a new-found freedom in the early stages of the second half and were giving their hosts plenty to think about on the break.
Some tenacious work from Anya allowed him to work his way into a crossing position and find Naismith at the back post, only for the Everton man to fluff his lines and fire over the bar.
Despite Scotland’s advances, the hosts still posed a threat in attack and Reus was denied by a sprawling Marshall save after Mario Gotze had caught out Hanley and threaded his former Borussia Dortmund team-mate through on goal.
Strachan introduced Steven Fletcher and James McArthur just before the hour mark and the Sunderland man almost made an instant impact as his near-post header was blocked in front of goal.
Fletcher did make a telling contribution on 66 minutes, though, as the ball found its way to his feet on a swift Scotland break. With his back to goal, the striker held up play before setting Anya away.
The winger, who had been lively throughout, raced through on goal and kept his composure brilliantly to slot beyond the onrushing Neuer and send the travelling fans into delirium.
However, Scotland were back on terms for little over four minutes as they immediately headed up the opposite end and Marshall was forced to push Gotze’s strike behind with two hands.
The visitors failed to clear the resulting corner as it bounced towards the six-yard box where the deadly Muller was lurking to steer his second of the evening into the roof of the net.
Strachan was quick to urge his side forward and McArthur could only poke a weak effort into the arms of Neuer after being found inside the area by late substitute Shaun Maloney.
A frantic finale looked to be in store as the fourth official signalled for three minutes of stoppage time, but the drama came inside the Scotland area as Gotze attempted to lob Marshall before Muller drilled a low strike against the post.
Charlie Mulgrew was then shown a second yellow card with seconds remaining as he kicked the ball away in frustration after conceding a free-kick, leaving the Scots to face the dying moments with 10 men.
Scotland’s next Group D qualifier is against Georgia at Ibrox on October 11, with a trip to Warsaw to face Poland following three days later.