Clark curled wide for Luton shortly after the restart, but from that moment on, it was all Stockport as they went in search of parity.
They went agonisingly close to a leveller when the ball dropped for Josh Stokes just inside the area, but his first-time effort went a matter of millimetres past the far post.
Stokes again nearly equalised, with a header at the near post from Ollie Norwood’s swinging free-kick – this time he was denied by an excellent reaction save by Shea.
With 10 minutes of stoppages added on, Stockport tried to force extra time, but Luton had other ideas.
Wells got across his marker from Shayden Morris’ ball into the box and it fell perfectly at his feet for him to provide a tidy finish to put the game beyond doubt.
Addai prevented Luton adding gloss to the scoreline, producing a wonderful save at full stretch to his left to stop Gideon Kodua’s goal-bound curling effort, but the damage had very much already been done.
Victory for Luton means they won the EFL Trophy for the second time, following on from success in 2009, while Stockport have now lost in the final of this competition three times.
Attention will soon turn to challenging for promotion from League One for both these sides.
But for now, after the highs of the Premier League, followed by a challenging two years with back-to-back relegations, it is Luton who can celebrate.