Billed as a clash of European heavyweights, the Germans ignored the soaring Salvador heat to deliver a sizzling attacking display against a disappointing Portugal, effectively ending the Group G game as a contest by halftime.
Portugal could not cope with the pace and movement of their much more organised and fitter-looking opponents and were 3-0 down by the interval as Mueller, marking his 50th international, scored either side of a thumping Mats Hummels header.
"Three goals in one match, in the opening match against such an opponent is just great," Mueller, whose side are bidding to end a 18-year title drought, told reporters. "We got into the match really well and had some half chances then suddenly we were ahead 2-0. In heat like this a lead like that is a great advantage."
Portugal coach Paulo Bento had feared the likely impact of the lunchtime heat, but his game plan failed to match the Germans' who went for a fast start before energy levels fell, with interchanging forwards Goetze, Mueller and Mesut Ozil causing havoc.
"We dominated the first half in the midfield with fast through balls to the strikers. We were clinical in finishing our chances, we had four goals from about six chances," Germany coach Joachim Loew told a news conference.
In their 100th World Cup match, the first team to reach the milestone, Germany were in the mood to party and went 1-0 up on 12 minutes through a Mueller penalty after the clever Mario Goetze wriggled away from Joao Pereira and was hauled down.
Hummels, who later limped off with an injury, powered home the second from a corner on 32 minutes before Mueller, top scorer at the 2010 World Cup, stole in ahead of a sleepy Bruno Alves to smash in the third in first-half stoppage-time.
The task was made even harder for Portugal by a needless 37th minute red card for Pepe. The defender, furious at what he felt was "simulation" by Mueller following a slight collision between the two, was sent off for pushing his head into the German's as he sat on the ground.
A poor qualifying campaign meant Portugal were forced to arrive in Brazil via a Cristiano Ronaldo-inspired playoff win over Sweden but their talisman and main goal threat could do little to impact proceedings at the Fonte Nova arena, cutting a frustrated figure throughout.
Much had been made of his inclusion after injury problems and although he began the day with wild screams meeting the mere mention of his name, jeers soon rang out as his infrequent touches came to little - a parried free kick his best effort.
With German Chancellor Angela Merkel watching in the crowd, it was Mueller who enjoyed the crowd's appreciation, completing the first hat-trick of the tournament after 78 minutes when he prodded home a spilled Andre Schuerrle cross. It was only the second hat-trick of the last three World Cups.
Portugal's best chance was a first-half Nani shot that narrowly fizzed over, while substitute striker Eder had a decent penalty shout denied after colliding with Benedikt Hoewedes.
Low points included goalkeeper Rui Patricio lazily presenting Sami Khedira with an open goal and Nani and Fabio Coentrao tackling each other on a rare foray into Germany's box.
The impressive victory extended Germany's run of having won every opening World Cup match since 1990 and they have scored at least four goals in five of those seven games.
For Portugal, who had only conceded once in their previous six World Cup group games, the defeat left qualification in the balance with games against the U.S. and Ghana to come.
"We have to analyse this in all calm," Bento said.
"For us the game was over by halftime and although we tried a bit in the second half, we didn't show much either. Germany controlled the game by holding the ball and the two decisions (penalty and red card) threw us out of balance."