The US, who lost the 2011 final to the Japan on penalties, booked their ticket to Sunday's championship game in Vancouver with a 2-0 win over top-ranked Germany on Tuesday.
Bassett's attempted sliding clearance from a low cross, crashed off the underside of the bar and over the goal-line to end England's dreams of a first final in the cruelest of fashions.
The teams had been set for extra-time after sharing two controversial first half penalties - Aya Miyama's 33rd minute spot-kick being cancelled out by Fara Williams' penalty seven minutes later.
England, playing in their first ever Women's World Cup semi-final, struck the bar twice in the second half and went close with several other opportunities but that counted for nothing after Bassett's mishap which left her distraught.
Defending champions Japan again left their victory to late - their quarter-final win over Australia had come with an 87th minute winner.