Sam Allardyce refused to commit his future to West Brom after suffering his first relegation from the Premier League following defeat at Arsenal.
The Baggies are the eighth club Allardyce has managed in the competition but a 3-1 loss at the Emirates Stadium saw them condemned to the drop.
The writing had been on the wall for some time and, despite a good start in London, goals from Emile Smith Rowe and Nicolas Pepe had them on the ropes.
Gunners boss Mikel Arteta said: “It is always the best way to put a remedy to the pain that we had – to win the game, to score the three beautiful goals that we have done.
“We created so many more chances and, as well, I think when we conceded the goal we looked nervy for a period – but overall, not going through to the final, it is always very important to win.
“I think we have a huge responsibility to win every game that we play when we represent this football club.
“And as well, if anybody can question anything – our level, our ability, whatever it is – they can, but our attitude is a non-negotiable and right up to the end we have to try win our matches and see what happens.”