Bayern Munich star Mueller bagged the first treble of his international career as Germany shrugged off a troubled build-up to the tournament to make a flying start in Group G.
But it was a disastrous afternoon at Salvador’s Fonte Nova Arena for Ronaldo and Portugal, who had defender Pepe sent off for a first half headbutt on Mueller.
Mueller opened the scoring from the penalty spot after 12 minutes before defender Mats Hummels doubled Germany’s advantage on the half-hour mark.
The 24-year-old centre forward struck again at the end of the first half before tapping home a close range effort on 78 minutes.
A wretched outing for Portugal was compounded by the sight of Real Madrid defender Fabio Coentrao being stretchered off in the second half.
But the biggest concern will be the form of Ronaldo, who had hoped to sparkle at the World Cup after helping Real Madrid to the Champions League this season.
The 29-year-old winger, who was battling injuries before the tournament, had declared himself 100 percent fit on the eve of the match.
However the former Manchester United star looked way below his best, and a poorly scuffed free-kick summed up his afternoon of disappointment.
In Group G’s other game on Monday, 2010 quarter-finalists Ghana face the United States in Natal having knocked out the Americans in the last 16 four years ago.
African champions Nigeria, who have taken just two points from six matches in their last two appearances at the World Cup, face three-time Asian champions Iran in Curitiba in Group F.
– Iran determined –
“Iran has faced difficulties off the pitch but we want to make our point on the pitch,” said Iran’s 61-year-old Portuguese coach Carlos Queiroz who cut his teeth as Alex Ferguson’s right-hand man at Manchester United.
“The economic and political situation is a complete disadvantage for the team but the players are very determined.”
Meanwhile Monday, FIFA chiefs said they would tweak the system of goal-line technology being used at the World Cup after controversy in Sunday’s game between France and Honduras.
Karim Benzema’s strike early in the second half of the Group E contest cannoned back off the inside of an upright across the goal at Porto Alegre’s Beira-Rio Stadium.
Honduras goalkeeper Noel Valladares’ tried to scoop it clear but the German-manufactured GoalControl goal-line technology system signalled the ball had crossed the line, prompting Brazilian referee Sandro Ricci to whistle for a score.
There was a brief delay as Ricci consulted his fourth official while Honduras manager Luis Fernando Suarez angrily protested but Valladares’s own-goal stood.
FIFA spokeswoman Delia Fischer said the “unique” nature of the incident — the ball striking a post and rebounding into the Honduran goalkeeper — had caused the confusion.
“Yesterday’s situation was quite unique,” she said, because it had triggered two animations.
“We are working to make it more clear. We will modify so you just see the ‘goal’ decision,” Fischer said.
“There were no mistakes yesterday. It all worked as planned. However we are working to modify it to make much clearer for fans inside the stadiums.”
Goal-line technology is being used for the first time at the 2014 World Cup following a U-turn by FIFA President Sepp Blatter.
It followed the notorious incident at the 2010 finals when England midfielder Frank Lampard had a valid goal disallowed against Germany despite replays showing the ball had crossed the line by several inches.