In the Group G match which marked a century of World Cup matches for Germany, the most for any country, the three-time champions destroyed Cristiano Ronaldo-led Portugal with Mueller leading a brilliant attacking brand of football with goals in the eighth, 45+1 and 78th minutes.
Defender Mats Hummels made his presence felt with a 32nd minute headed goal in an afternoon kick-off match which saw Portugal centre back Pepe being sent off in the 37th minute after an ill-tempered tangle with Mueller.
The match at Arena Fonte Nova, however, did not live up to the billing of a marquee clash with Germany toying with their opponents and then made worse for Portugal after Pepe's sending off.
Ballon d'Or winner Ronaldo, who made it to the Portugal playing eleven after being bothered by a left-knee injury in the build-up, made little impact in the game. He threatened early in the match, including a shot from a break that Germany goalkeeper Manuel Neuer could only block. His other significant involvement was a fine free kick late into the match but Neuer was upto the task.
Germany now placed themselves in a strong position to advance to the round of 16 from the so-called Group of Death which also has Ghana and United States.
Pepe, who has a history of ill-tempered misconduct, will now miss Portugal's next two matches which will surely hurt the Iberian country in their chances for a knock-out round berth.
Germany coach Joachim Loew deployed three attacking midfielders without a classical striker and used captain Philip Lahm as defensive midfielder instead of his favourite right back position where Jerome Boateng was tasked to try and stop Ronaldo.
The ploy worked with Germany running over their opponents and keeping Ronaldo more or less quite.
Playing in his 50th for Germany, 24-year-old Mueller opened the scoring after Mario Goetze was pulled down by Joao Pereira in the box, coolly slotting the resultant penalty kick into the right corner of Rui Patricio's goal.
Mueller received applause from Germany Chancellor Angela Merkel who stood up and clapped in appreciation.
Hummels then rose above Pepe to double Germany's lead with a well-timed header from a corner.
Pepe was sent off in the 37th minute for charging on and apparently head-butting Mueller who went down theatrically after being hit on the face by his rival.
Mueller scored again just before the break, stripping the ball from Bruno Alves and drilling in a low shot. He completed his hat-trick in the 78th minute in a goal-mouth scramble after Portugal goalie Patricio had parried a shot.
In the process, Mueller, who scored five goals in the 2010 finals in South Africa to claim both the Golden Boot and best young player award, scored his 20th international goal and eighth World Cup goal over two editions.
With today's win, Germany continued their domination of Portugal by adding Brazil 2014 to their list of recent wins over Ronaldo's side which includes the 2006 World Cup, plus the 2008 and 2012 European championships.