The Tigers, relegated from the top flight on the final weekend of last season, created the better chances across the 90 minutes but struggled to put the ball in the net.
Just six minutes had transpired before the first shot of the match arrived, as Tom Huddlestone was allowed to put his foot through the ball 20 yards from goal only to send his shot right down the throat of Westwood.
The opening stages of the match remained balanced, but Wednesday were arguably the quicker of the two sides to settle and managed to fire in two shots on goal through Fernando Forestieri and Ross Wallace.
Both attempts came from difficult shooting positions, with Wednesday taking a shoot-on-sight policy up against stand-in City stopper Eldin Jakupovic, who once more had to deputise for the injured Allan McGregor.
Hull well and truly took control of the second quarter of the contest, however, coming close on a number of occasions as their tag of pre-match favourites began to show all over the field.
Abel Hernandez was only denied goal number 24 of the season by a Tom Lees block on the line, yet the Tigers felt they should have had a penalty when the follow-up attempt brushed against the midfielder's hand.
It almost mattered little moments later when eventual matchwinner Diame burst up the field into the box, even getting a shot away at the end of the move with three opposition players surrounding him.
Diame's shot appeared to be a little scuffed, but he got the better of Westwood at his front post only for the frame of the goal to this time come to the rescue of Carlos Carvalhal and his side.
Hernandez and Moses Odubajo both saw big chances of their own come and go in the latter stages of the first half, with an inspired Westwood proving to be the difference by keeping out both shots with the full use of his large frame.
All off Hull's hard work would have counted for nothing had Lees planted his header either side of Jakupovic from the final act of the opening 45 minutes, as the two sides went into the break all square - just as they did in both regular league meetings.
Wednesday, who last visited Wembley three decades ago, again came out of the blocks quicker following the restart and saw a big opening fall their way when Forestieri pounced on a Michael Dawson slip.
Central-defensive partner Curtis Davies got across well on that occasion to deny the 15-goal ace a shot on target, but it was a familiar story at the other end soon after when Westwood was called into action to thwart Dawson.
Andrew Robertson was the latest Hull player to fluff his lines when picked out unmarked by Odubajo just a few yards from goal, getting under the ball and blasting high into the stands.
Both teams pushed hard for a way through, knowing the importance of the first goal in this fixture down the years, although the chances began to dry up as normal time wore on.
Then, just when a moment of magic was required, Diame popped up to curl a shot into the back of the net from range which Westwood's fingertips could not direct over the bar on this occasion.
Only one side in the last 17 seasons, Queens Park Rangers in 2014, had finished fourth in the end-of-season table and progressed all the way through the playoffs - a figure the Humberside outfit were happy to add to as they saw the game through, despite their opponents ending the game with four strikers on the pitch.