Muller's double secured all three points for Joachim Low's side in Dortmund on Sunday but the world champions were made to work hard in their opening Group D clash.
The forward – who scored five goals at this year's World Cup – gave Germany the lead with a looping header that he later acknowledged was not intentional.
Ikechi Anya drew Scotland level in the 66th minute but parity lasted just four minutes before Muller struck again.
Muller said: "Making it 2-1 was a joy, because I know how important those three points are.
"I could have scored the opener with a header earlier.
"Football is football – you try your best, but I did not intend to place the header [for the opening goal] where it went.
"I just tried to win the aerial challenge against this towering Scotsman [Russell Martin]."
Germany defender Benedikt Howedes felt his side deserved their win on the basis of their first-half dominance.
"It was hard work today – a difficult win," Howedes said.
"But against a strong Scottish side, that was no less than we expected.
"The first half was a one-way street. In the second half, they got a bit braver, invested more in attacking and equalised.
"It was good that we made it 2-1 shortly afterwards."
