On his return from a back injury, Woods (75) finished at seven over – three shots below the cut – at the Congressional Country Club.
Australian duo Leishman (66) and Goss (66) surged into a four-way share of the lead at the PGA Tour event, which is hosted by two-time champion Woods.
American pair Ricky Barnes (69) and Patrick Reed (68) are also at six under in Maryland.
Leishman, who opened with a 70, went through his round bogey-free and holed birdies at the first, sixth, ninth, 11th and 15th.
Goss, 20, also went through without a blemish as he moved into contention for his maiden PGA Tour win.
Barnes was left to lament back-to-back bogeys at holes four and five, while Reed double-bogeyed the 11th.
American Hudson Swafford (68) and Australian Stuart Appleby (67) are tied for fifth at five under after solid rounds.
Swede Freddie Jacobson (71) mixed three birdies with as many bogeys to be equal seventh with three Americans.
Morgan Hoffmann (68), Billy Horschel (68) and George McNeill (69) are alongside Jacobson after good rounds.
Woods, who insisted his injury is no longer troubling him, produced a double bogey, five bogeys and three birdies.
Overnight leader Greg Chalmers of Australia had a poor round of seven-over 78, including a double bogey at the third hole, to be at two over and tied for 50th.
Defending champion Bill Haas (72) is tied for 13th at two under.
Among others to have missed the cut were world number six Jason Day and South African Ernie Els.