Germany 2 Scotland 1: Muller double downs plucky visitors
A 4-2 friendly defeat to Argentina on Wednesday raised alarms in the Germany camp and, although Joachim Low's men were far from their best in Dortmund, an improved performance earned the hosts three points.
Muller headed Germany in front in the 18th minute of a first half that they dominated, but Scotland greatly improved after the interval and levelled through Ikechi Anya.
The visitors' joy lasted just four minutes, though, as Muller's scrappy second – bundled home from five yards after Scotland failed to clear a corner – proved enough to see off Gordon Strachan's brave side, who had Charlie Mulgrew sent off for a second yellow card in second-half stoppage time.
Now without retired trio Philipp Lahm, Miroslav Klose and Per Mertesacker, Germany have much room for improvement, and they will soon be boosted by the returns from injury of new captain Bastian Schweinsteiger, Mesut Ozil and Sami Khedira.
Marco Reus' late ankle injury may put a dampener on Germany's win, but the fact they withstood a bout of second-half pressure to prevail will undoubtedly please Low.
Muller and World Cup hero Mario Gotze headlined Germany's four changes while Scotland captain Darren Fletcher made his return to international football, following ulcerative colitis that had kept him out since October 2012.
David Marshall was preferred to Allan McGregor in goal for Scotland and he would have been relieved to see an unmarked Muller nod wide from eight yards early on.
Muller then had a header cleared by Russell Martin in the ninth minute and while Germany controlled the early stages, they were given a warning by Mulgrew, who rattled the crossbar from an offside position shortly after Barry Bannan's effort had been deflected wide.
Marshall was called upon to turn an Erik Durm effort around the post in the 16th minute but he could do nothing about Muller's opener.
The Bayern Munich forward – who scored five goals at the World Cup – was sandwiched in between Martin and Alan Hutton but rose highest to loop a header from Sebastian Rudy's chipped cross over Marshall and in.
The goal triggered a spell of German pressure; Reus and Rudy both shooting at Marshall, while Martin's well-timed block kept out Andre Schurrle and Grant Hanley almost turned a cross into his own net, with his goalkeeper sparing his blushes.
Anya caused occasional problems for the home side with his pace but hurriedly hit an effort straight at Manuel Neuer on the stroke of half-time and Steven Naismith was denied by the outside of the post when play resumed.
Hutton burst down the right to feed Naismith, who beat Jerome Boateng and Benedikt Howedes, but his shot glanced agonisingly off the upright with Neuer beaten.
An unmarked Naismith then missed the target from 12 yards as Strachan's charges continued to press and, just after Marshall had kept out Reus, the visitors got their equaliser when Steven Fletcher found Anya in acres of space.
The Watford man stormed forward and held his nerve, coolly slotting past Neuer to level proceedings with 24 minutes left.
Germany's response was swift, though, as Muller's instinct around goal proved decisive.
The goal still did not deter Scotland, with Naismith hauled down by Durm in a good position and James McArthur screwing a shot at Neuer.
But Germany, who saw Muller hit the post in the dying stages, did enough to hang on for victory - albeit one that may be soured by Reus' late ankle injury - as Mulgrew was given his marching orders for kicking the ball away.