The two games helped the World Cup continue to deliver lots of goals — 44 in the first 14 games.
Nigeria played Iran in the day's other match, and neither team added to that tally. The teams played the first draw of this World Cup, finishing scoreless. Argentina defeated Bosnia 2-1 on Sunday in the other Group F match, and both will feel confident after watching Nigeria and Iran fail to produce.
Two seeded teams — defending champion Spain and Uruguay — were upset earlier. That fate would not befall Germany on Monday.
It was Germans' 100th World Cup match, and they had it wrapped up by halftime.
Mueller, who led the 2010 World Cup with five goals, scored on either side of Mats Hummels' headed goal as Germany took a 3-0 lead at the break. He added his third goal in the 81st.
"To score three goals in the World Cup opener against such an opponent is great," Mueller said.
"After 20 minutes, we looked up at the clock and thought it was going to be a long day," he added. "But we got into the match very well and when you lead by 2-0 in this heat and then even get the third, it was all over."
Cristiano Ronaldo started at Arena Fonte Nova despite some nagging injuries. The winner of FIFA's best player award, he faded after threatening early in the match. He had a shot from a break that Germany goalkeeper Manuel Neuer blocked, but there was nobody from Portugal to pounce on the rebound.
The United States scored at beginning and the end, but for most the match it was on the back foot as the Africans attacked, pushed the Americans deep into the box, but failed to finish in a physical match.
Dempsey's goal was the quickest ever by an American and one of the fastest in World Cup history. He's the first American to score in three different World Cups.
Brooks got the winner on a header in the 86th, just four minutes after Andre Ayew equalized.
The 21-year-old defender was a second-half substitute and an unexpected addition to the American roster. He was overcome and fell to the ground as teammates piled on. He made his national team debut in August.
"I said it to the bench minutes before, we're going to get some chances still," U.S. coach Jurgen Klinsmann said. "So we are still in the game after the equalizer, we just need to kind of push and push and grind it out. That's what they did."
U.S. players ran onto the field to celebrate at the final whistle, jumping as supporters chanted "U-S-A! U-S-A!"
"We have a great spirit," Klinsmann said. "The U.S. team always has a great spirit and fights until the last second."
The defending Cup of Africa Nations champion, Nigeria will be more disappointed with the draw than Iran.
Neither Iran nor Nigeria has won a match at the World Cup since 1998.