Wambach, playing in her 98th international, scored both goals for the Americans to rip the heart out of the Swedes who are gunning for their first World Cup title after finishing runner-up in 2003.
The victory means the United States, two-time champions and world number ones, have hit the top of Group B and are almost certain of reaching the quarter-finals, with only Nigeria left to play at the group stage.
Both sides had been desperate to win on Friday and secure the three points after drawing their opening encounters in a group considered the toughest of the tournament.
The US drew 2-2 with Asian number ones North Korea earlier this week, while Sweden drew 1-1 with Nigeria, seen as the weakest of the four teams.
Wambach, who suffered a bloodied head in the first match that needed stitches, struck in the 34th minute by coolly converting a penalty past goal goalkeeper Hedvig Lindahl.
Swedish defender Stina Segerström had seconds earlier brought down Lori Chalupny as they desperately scrambled for a long ball metres from the goal.
Hungarian referee Gyoengyi Gaal had no hesitation in handing the distraught Segerström a yellow card and pointing to the spot.
Wambach then nailed victory for the Americans by chesting down a pinpoint pass inside the box in the 58th minute and unleashing a bullet from her right foot past a diving Lindahl.
As the minutes ticked down, the Swedes launched a wave of counter attacks, spearheaded by tenacious forward Hanna Ljungberg, playing in her 122nd international.
Ljungberg, who once received an invitation to play for the Serie A men’s side Perugia but turned it down, almost scored in the 79th minute when she connected with a cross but her shot went wide.
The Swedes kept coming but were unable to find a way through the solid American defence, superbly marshalled by Cat Whitehill and veteran Christie Rampone.