Having gone 2-0 up through goals from Greg Pearson and Sam Wright in the first half, Hammers were pegged back as Middlesbrough fought back to draw level in the second half.
Pearson struck what looked to be the winner in the 83rd minute, but Boro - who included many of the team that reached the FA Youth Cup final last season - snatched an equaliser in the final minute to make it honours even.
Said Tony Carr afterwards: "I would have to say it was a fair result in the end. We were quite fortunate to come in 2-0 up at half-time and they dominated much of the second half, so a draw was probably right.
"Having said that, it was still disappointing to concede a goal so late in the game and it would have been a real boost for us to start with a win against a very decent side.
"We played some good football, though, and the players gave their all, so I can't have any complaints."
Team: Behcet, McClenahan, Cohen, Lumsden, Blewitt, Wright, Tucker (Laws), M Noble (Collington), Pearson, Bunce, Cole (Parrington).