Stuart Pearce's side were a matter of minutes from qualification for the semi-finals after Welbeck had put England ahead in their final group game against the Czech Republic.
But the Czechs drew level in the 89th minute through Jan Chramosta, before Tomas Pekhart netted a winner in injury time to send them through and put England out.
"I don't know how we're going out after that performance," said a stunned Welbeck afterwards.
"All the lads have taken a hit and it's just a down feeling. We had the tactics right, keep the ball moving and hopefully wait for our chance - we took it, and were just unlucky to lose the game in the end."