James was fouled by Willie Green as he attempted a 3-pointer.
"I'll take it," James said. "I feel real good out there. I just tried to make a play and got the call. It went our way, which is a good feeling."
It was a fairly controversial ending and left further questioning about the replacement referees. Reports are circulating that the locked-out referees could have an agreement to return in time for the regular season.
"I didn't think it was a foul," Green said. "From what he saw, he called it. These guys have a tough job and they're doing the best they can. That call didn't lose the game for us."
The 76ers appeared to have the win wrapped up when Green hit a free throw with 1 second remaining. On Philadelphia's final possession, Green pump-faked Young and drew a foul. Green missed the first attempt and made the second for the one-point lead.
Washington inbounded at midcourt and James, who finished with 15 points, was able to get off the shot.
A sparse crowd attended the game with patches of empty pockets of seats in the lower level. The final Bruce Springsteen concert ever to be held across the street at the Spectrum may have played a role.
A dunk by Smith put the Sixers ahead 77-75 with 6:02 remaining, but the Wizards reeled off a 9-0 run to take an 84-77 lead with 4:06 left.
Philadelphia went on an 11-2 run to and went ahead 88-86 when Speights hit a short hook shot with 43.8 seconds remaining.
Young followed with a 15-foot jumper to tie the game at 88 with 31.9 seconds left.
The game was ragged throughout the first half with Washington taking a 39-34 halftime lead.
The teams combined for 19 turnovers and 54 missed field goals in the first half. ... F Rodney Carney (left hamstring strain) missed the game for the Sixers.