Honduras, third in the six-nation final qualifying group in North and Central America and the Caribbean region behind Brazil-bound USA and Costa Rica, took their points total to 14 with one match remaining.
Mexico and Panama went into their match at Azteca Stadium on eight points each.
A draw between them would put Honduras through, and leave them fighting to finish fourth. The fourth-placed team will play off for a World Cup berth against Oceania's New Zealand.
In a match that produced few clear-cut scoring chances, Bengtson grabbed the only goal Honduras needed, slotting home a shot from the center of the area after taking a smooth pass from Wigan midfielder Roger Espinoza.
The goal sparked joyous celebrations by fans in the stands at Estadio Metropolitano, where Brazil banners waved alongside Honduran flags.
Honduras' Colombian manager Jose Luis Suarez was circumspect, however, with the Mexico-Panama match still to come on Friday, calling the victory just "one more step" in his team's campaign to reach a second straight World Cup.
Graham Zusi scored in the 77th minute and Jozy Altidore doubled the score in the 80th as the Reggae Boyz's slim hopes of making a play-off were extinguished.