The Germans got off the mark after just 11 minutes as Muller converted a penalty kick, and the advantage was doubled in the 32nd minute through a goal from defender Mats Hummels.
Things got worse for Portugal as defender Pepe was shown a straight red card following an altercation with Muller, and the Bayern Munich man took full advantage in first-half stoppage time when he buried his second goal of the day.
Muller finished his treble in the 78th minute when he fired one home from close range to become the fifth German player to score a hat trick in the World Cup.
Germany surges to the top of Group G, while Monday's other contest in the group sees the United States kick off its World Cup journey against Ghana in Natal.
More to follow ...