The Germans delivered a sizzling attacking display that underlined their tag as strong title contenders and the match, billed as a clash of European heavyweights, was over as a contest by halftime.
Two Mueller goals and a thumping Mats Hummels header blew away a Portugal side whose misery was compounded 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 face as he sat on the ground.
In their 100th World Cup match, the first team to reach the milestone, Germany were in the mood to party in the sapping Salvador sunshine and went ahead after 12 minutes through Mueller from the penalty spot after the clever Mario Goetze was hauled down.
Hummels powered home the second from a corner on 32 minutes before Mueller, top scorer at the World Cup four years ago, stole in ahead of a sleepy Bruno Alves to smash in the third in first-half stoppage-time.
The interchanging attack of Goetze, Mueller and Mesut Ozil bamboozled Portugal, whose talisman Cristiano Ronaldo was left a frustrated observer for much of the match, a late long-range free kick parried by Manuel Neuer his best moment.
With German Chancellor Angela Merkel watching in the crowd, Mueller completed the first hat-trick of the tournament after 78 minutes, prodding home a spilled Andre Schuerrle cross and extending his country's run of having won every opening World Cup match since 1990.
The demolition will send a strong message to Germany's rival title contenders in Brazil and left Portugal looking nervously at qualification with games against the U.S. and Ghana to come.