Sheldon, an 88th-minute substitute, lit up a frantic extra period at Glanford Park with fantastic strikes after 92 and 102 minutes. His brace of goals, added to Andrew Dawson's second-minute opener, gave Scunthorpe a 3-2 aggregate win.
The Swans were desperately unlucky, though, They twice hit the woodwork in normal time before Tony Bird levelled the aggregate score at 2-2 after 98 minutes.
The 102nd-minute dismissal of Scunthorpe's John Eyre ensured a hectic ending, and Swansea's Jason Price missed an open goal late in extra time.
In the other semi-final second leg at Millmoor, Orient overcame the home side, Rotherham United, 4-2 in a penalty shoot-out, after both sides had failed to score in three and a half hours of football. The Orient goalkeeper, Scott Barrett, saved two spot-kicks in the decisive shoot- out.