Felix Nmecha opened the scoring after just six minutes, finishing off a slick move involving Florian Wirtz. Curacao, the smallest nation ever to compete at a World Cup, briefly threatened an upset when Livano Comenencia levelled in the 21st minute with a composed finish from inside the box.
Germany regained control before the break through Nico Schlotterbeck, who headed in from a Brown corner, before Havertz converted a penalty in stoppage time to make it 3-1 at half-time.
The second half belonged entirely to the four-time champions. Jamal Musiala restored a two-goal cushion early on, before Brown got his goal in the 68th minute. Substitute Deniz Undav, who had been criticised by his own head coach before the tournament, added a seventh with a goal of his own after setting up Brown moments earlier. Havertz completed his brace in the 88th minute.
Speaking after the match, head coach Julian Nagelsmann said the scoreline was exactly what his squad needed heading into a demanding group.
"It took us a few minutes to get back into the game after they equalised. Curacao can play football too, as we saw, and I'm curious to see how they'll fare in the group going forward."
He went on to praise the manner of the performance overall.
"I'm very satisfied with us scoring seven goals and our performance for the most part. A winning start is always important and we're glad we managed it."
Brown, the 22-year-old Eintracht Frankfurt left-back making his World Cup debut, was among the standout performers and spoke about the significance of the occasion afterwards.
"It's indescribable that I scored in my first World Cup match. We're very pleased to have scored seven goals."
He acknowledged that the goal conceded by Germany was avoidable, but suggested it may have sharpened the team's focus.
"The goal we conceded was unnecessary, but perhaps it did us good to wake us up. I'm proud of the team for simply carrying on and playing well. The goals are good for us, but we also need to keep working on our weaknesses."
Germany next face Ivory Coast, a side Nagelsmann has previously described as one that should not be taken lightly despite being considered beatable on paper. Ecuador complete Group E, a team the head coach has called difficult to play against given the quality in their attacking ranks.
For a Germany side that failed to escape the group stage at the last two World Cups, the manner of Sunday's win offers genuine encouragement. Seven goals against any opposition sends a message, but it is the response after Curacao's leveller that Nagelsmann will point to when tougher tests arrive.