Wasps poked their noses in front for the first time three minutes into the second half as Hughes showed all his mighty strength and power to shrug off a posse of Chiefs rivals and spark a surge down field.
Exeter eventually regrouped and cut the deficit to three but looked to have paid a heavy price for twice opting for a scrum when presented with kickable penalties - until they got a third a minute from time which Steenson landed to take the game to extra time.
First-half tries by Daly's Lions colleague Jack Nowell and full-back Phil Dollman - plus two Steenson conversions - built a 14-3 advantage, yet Gopperth's touchdown just before half-time appeared to tilt the balance psychologically.
Jimmy Gopperth's 15 points were not enough for Wasps to win the English title.
Cipriani made the original break, fed to hooker Tommy Taylor who combined with Dan Robson to send Gopperth darting through and over for a try he converted himself.
However, the Chiefs finally saved their own skin, forcing extra-time before a Gareth Steenson penalty in the dying moments secured a 23-20 victory.
But then Wasps collapsed a scrum and referee JP Doyle gave Steenson the easy task of sending his kick over.
Hero Steenson said: "It was such a tough game with the emotional highs and lows. It is just another 80 minutes of rugby in which we are determined to do ourselves justice and to produce a performance we can all be proud of".
"The final is new territory for this group of players, but really exciting territory and something our players and coaches can't wait to be involved in", Young said.
"To be standing here from where we were seven years ago is absolutely incredible".
"One side are always going to come out on the wrong side and unfortunately that is us today".