The Hammers led twice and looked to be heading for a third straight win when Patrick Kisnorbo's own-goal made it 2-1 in the second half.
But as the clock ticked down Leeds, who earlier missed a penalty through Max Gradel, snatched an equaliser through 22-year-old midfielder Clayton.
Sam Allardyce's side had previously taken the lead in the sixth minute at Upton Park when Carlton Cole found the net while Leeds' first equaliser came after the break from Ross McCormack.
Faris Haroun came off the bench to score on his debut as Middlesbrough continued their impressive start to the season with a 3-1 win over Birmingham at the Riverside.
The Belgium midfielder, introduced at the start of the second half, fired Boro 2-1 ahead on 69 minutes before fellow substitute Malaury Martin put the game out of reach with a third.
Adam Rooney had given Birmingham the lead from the penalty spot, only for Barry Robson to level just after the restart.
Boro continue their unbeaten start to the season with their third consecutive victory.