Higuain then capitalized on a blunder by Mexico defender Ricardo Osorio who passed the ball straight to Higuain just outside the penalty box. The Argentinean striker calmly rounded goalkeeper Oscar Perez before slotting into an empty net in the 33rd.
Tevez arguably scored the goal of the tournament so far on 52 minutes by smashing an unstoppable shot into the top-right corner just outside the penalty area.
Mexico forward Javier Hernandez earned his side a consolation goal when he fired a shot past Argentina goalkeeper Sergio Romero in the 71st minute.
Germany powered into the last eight by crushing England 4-1 in a match that will be remembered for the goal not awarded to Lampard.
England coach Fabio Capello believed that referee Jorge Larrionda's wrong decision was the turning point and eventually caused a disastrous end to his team's South African journey.
"If we had been awarded the goal at 2-2, the game would be different," said the Italian.
"It's a big mistake from the referee. It's too important for us to go forward," he added.
Thomas Mueller finished two quick counterattacks within four minutes in the second half to sink England's hopes of beating Germany at the World Cup for the first time since the 1966 final and send the English to their worst ever defeat at a final tournament.
Miroslav Klose broke the deadlock after 20 minutes for Germany and opened the goal flush in a highly-anticipated match.
Lukas Podolski doubled the Germans' lead in the 32nd minute. Although England pulled one back only five minutes later through Upston's header, Thomas Mueller made a double within three minutes after the break to put the victory beyond the reach of Capello's squad.
Seven minutes before halftime, Lampard's blasting shot bounced off the underside of the bar and landed about half a meter over the goal-line.