Three-time champions Germany celebrated their record 100th World Cup match, the most for any country, in style by steamrolling Cristiano Ronaldo-led Portugal who played for nearly an hour with 10 men after defender Pepe was sent off for attempting a head-butt on Mueller in the 37th minute.
Mueller led a brilliant attacking brand of football with goals in the 12th, 45+1 and 78th minutes while defender Mats Hummels scored the other one in the 32nd minute.
Ballon d'Or winner Ronaldo, who made it to Portugal starting line-up after being bothered by a left-knee injury in the build-up, made little impact on the game. He threatened early with 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.
Elsewhere in Curitiba, Nigeria and Iran delivered the first draw of the World Cup on Monday as they ground out a scrappy 0-0 stalemate in their opening match in Group F.
The draw at the Arena da Baixada in the southern Brazilian city of Curitiba followed 12 mostly high scoring and attacking games.