Mexico's unhappy evening was complete when Rafael Marquez -- dismissed in their 2002 World Cup defeat by the same opponents -- was sent off in the 66th minute.
The win extended the United States' recent dominance over their neighbours, who have failed to win in their last five meetings.
The match kicked off the six-team final stage of the CONCACAF qualifying tournament and the defeat was a huge blow to Mexico, who also lost in Honduras and Jamaica in the previous stage of the competition.
Eriksson has already faced media calls to quit and the result will heighten speculation over whether Mexico can even qualify for the next World Cup in 2010.
El Salvador, Honduras, Costa Rica and Trinidad and Tobago complete the group. The top three teams qualify for South Africa and the fourth plays off against the fifth-placed South American side for another place.
The United States dominated the match in conditions made difficult by strong wind and heavy rain before kick off.
Landon Donovan threatened with a free kick that hit the Mexican wall and went narrowly wide, and Mexico goalkeeper Oswaldo Sanchez made a fine close-range save on the half hour to deny Clint Dempsey.
Mexico then lost striker Nery Castillo, who limped off injured and was replaced by Israel Martinez in the 35th minute.
The United States went ahead in the 43rd minute when Bradley snapped up a rebound after Sanchez blocked Oguchi Onyewu's shot.
Mexico tried to muster a more attacking edge after halftime but rarely looked dangerous and were reduced to 10 men when Marquez lashed out at U.S. goalkeeper Tim Howard as the two went for a high ball.
Bradley added the second for the United States in injury time with a long-range shot that slipped under Sanchez's arms to leave the Mexicans facing another angry reception on their return home.
