Muller scored from the spot in the 11th minute, before Mats Hummels doubled the lead with a thumping header from a corner.
Pepe then saw red for a coming together with Muller, which made it comfortable for Germany and the Bayern Munich man to pick them off with two further goals.
Here, Sports Mole takes a closer look at the action.
On target: 9
On target: 9
Was the result fair?
Absolutely. Even before Portugal centre-back Pepe was dismissed for pushing his head into Muller, Germany were well in control. Portugal were sloppy in possession in the opening 10 minutes and the German movement, after a slow start, was having its say. It is quite surprising that the shot and possession statistics are so close, but Germany had by far the better efforts, with the Portuguese limited mainly to strikes from range and half-chances. Germany were clinical, but they did completely outclass their opponents.
They took a while to get going, with the likes of Philipp Lahm unusually sloppy in possession early on. However, once they did start to find feet with passes there was only one likely winner. Mario Gotze started as the focal point of the attack but he, Muller and Mesut Ozil had a very good understanding and they took up each other's positions throughout with ease. Portugal gave away a number of fouls, and that represents how good the German movement was in this game.
There was not one player who stood out for Portugal in a performance that was both sloppy and scrappy. Cristiano Ronaldo's teammates seemed too keen to pass the ball to him that on a couple of occasions they even gave up a good shooting chance to find him. There were one or two decent performances, including from Bruno Alves, but the rest were far too inconsistent and by the time their attack got going in the second half the match was long over as a contest.
Sports Mole's man of the match
Thomas Muller: Of course this award will go to the hat-trick hero Muller. A lot of credit must also go to Gotze, who shares what seems to be a telepathic understanding with his teammate, but Muller troubled every one of the Portuguese defenders at various points in the game. His initial reaction to a slight touch from Pepe's arm was disappointing, but that should not detract too much from what was a match-winning performance. He has eight goals in seven World Cup games now and looks good to be top scorer.
With the score at 0-0, the shaky-throughout goalkeeper Rui Patricio scuffed a clearance from the left side of his goal straight to Sami Khedira. The Real Madrid man took on the shot first time from 40 yards but, fortunately for the Portugal stopper, put his effort wide. It would have been the big mistake of the tournament so far.
Milorad Mazic had quite a few big calls to make today. He was spot on to award the early penalty to Germany, and his decision not to send off Joao Pereira was brave but sensible. The red card for Pepe was also correct, but there might have been a penalty for Portugal as Eder looked to be fouled by Benedikt Howedes. Overall, however, Mazic did well in a tough game to referee.
Germany: The group favourites march on to take on Ghana on Saturday, where a positive result could see them already into the knockout stages.
Portugal: The losers must dust themselves down quickly for their clash with USA on the same day. It has become, really, a must-win game against a team that could now fancy their chances of an upset.