Assistant boss Fletcher watched his charges twice come from behind at the Galpharm Stadium before bowing out 4-2 on spot-kicks.
Lee Peltier put the hosts in front before Steve Lovell equalised from the penalty spot after Adam Smith had been brought down by Peter Clarke.
Danny Ward, though, restored the Terriers' lead just two minutes after Lovell's penalty.
Lovell then equalised for a second time with a finely-taken effort just after the hour to force extra-time.
Danny Ings headed Cherries into the lead for the first time in the first period of extra-time, only for Antony Kay to make it 3-3 moments later.
Jason Pearce was sent off for a two-footed lunge on Kevin Kilbane, before Liam Feeney and Anton Robinson both missed their shoot-out kicks.
An emotional Fletcher, who shed tears after the Terriers' triumph, said: "The first thing that comes to mind is pride. I'm so proud of what we have done as a team and where we have come from.
"Going back two-and-a-half years when I came back to the club, what the majority of those boys have been through is more than you usually get in a whole career.
"Proud is the only word I can use to describe how I feel, but I'm absolutely deflated.
"The one thing missing off my CV is playing in the Championship and I thought this was going to be it. I kept looking up to the Gods and praying, but it just wasn't to be.
"I have watched many play-off semi-finals on TV and thought how awful it must feel to lose on penalties, but it doesn't prepare you for actually being involved in it.
"You can multiply by a thousand how awful it is, but I thought our boys were magnificent - I really did."
FULL MATCH REPORT IN TOMORROW'S ECHO WITH EXTENDED REACTION IN FRIDAY'S EDITION.