The underdogs claim the trophy after winning both legs, an own goal from Adam Nditi and strikes from Cameron McGeehan and Joshua Murphy adding to the 1-0 advantage from Carrow Road
In an action-packed first half, Boga levelled the aggregate score with a smart finish before an own goal from Nditi, swiftly followed by a McGeehan penalty - converted after Carlton Morris was brought down - put the Canaries in the driving seat going into the second half.
The Blues attempted to launch a comeback but, in spite of concerted periods of pressure, they struggled to break down a stoic Norwich defence as the away side held out to claim an unexpected win, even adding to their tally with a late Joshua Murphy tap-in before Boga grabbed his second of the evening - a goal that proved scant consolation for the home side.
Chelsea, without their most senior figure in Nathan Ake due to his involvement in Tuesday's Europa League final squad, began the match in the ascendancy and made their early dominance pay with just under 15 minutes gone. The dangerous Boga picked up a pass from Andreas Christensen before turning to run at the retreating Norwich defence, eventually firing a shot past the despairing William Britt and into the bottom corner.
Norwich reacted swiftly, though, as it took the away side just seven minutes to find a response; winger Joshua Murphy picked up a short corner, catching the Blues completely unawares as he worked his way to the byeline before driving in a dangerous cross. Nditi, the Chelsea left-back, got an unfortunate touch on the ball at the near post, diverting the ball past Mitchell Beeney and into his own net, levelling the score with 22 minutes gone.
The Blues quickly reclaimed the initiative following the equaliser, putting Norwich under significant pressure as Alex Kiwomya, John Swift and Islam Feruz all spurned chances over the course of a dominant 10-minute spell.
The Canaries soon capitalised on the home side's attacking intent, countering well and almost putting themselves further ahead as Carlton Morris forced the impressive Baker to clear off the line after he had turned Aina and fired past Beeney in the Chelsea goal.
With Chelsea continuing to attack, it was not long before Norwich took full advantage and it was Morris who produced the goods once more, first dispossessing Blues centre-back Alex Davey before then being brought down by the same man in the penalty area. Captain McGeehan duly converted the penalty, firing past Beeney to net his second penalty over the course of the two legs.
The Blues persisted in their attempts to overturn the two-goal deficit after the break but they struggled to find a way past a resolute Norwich defence, perhaps their best chance falling to Feruz as the striker spurned another golden opportunity after Swift put him through, rounding the keeper smartly but striking wide of the post.
The frantic pace Chelsea maintained in the first half appeared to tell in the closing stages as Chelsea proved unable to break down Norwich, before matters were made even worse by Joshua Murphy as, after a swift counterattack took the home side apart, his twin brother Jacob Murphy squared the ball for him to tuck home with consummate ease.
The Blues grabbed a late consolation goal courtesy of Boga, who grabbed his second of the goal after a well-worked move involving substitute Charly Musonda, though it proved too little too late for the home side.
Norwich: Britt; Norman, McFadden, Toffolo, Wyatt; McGeehan, Randall (Hodd 90+4); Jacob Murphy, King (Young 81), Josh Murphy; Morris.
Subs not used: Lokko, Reading, Brown.
Chelsea: Beeney; Aina (Musonda 72), Davey, Christensen, Nditi (Wright 69); Baker, Loftus-Cheek; Swift (Colkett 63), Feruz, Boga. Kiwomya.
Subs not used: Conroy, Killip.