The Toffees lost 2-1 to Manchester United on Saturday.
The guts of 32,000 Toffees supporters were in attendance as Roberto Martinez's side conceded a stoppage-time goal from Anthony Martial - the latest low point in a season littered with them from an Everton perspective.
Former Evertonian Marouane Fellaini had given the Red Devils a 1-0 half-time lead, before a Chris Smalling own goal on 76 minutes looked like being enough to force extra time, but it wasn't.
The FA Cup was the club's last chance for a successful season, with the Blues languishing 11th in the Premier League, having had very little to play for on that front for a couple of months now.
Cleverley admits that the result "really hurts" and is sad that he and his team-mates could not reward the fans' passion.
“The fans were magnificent and really got behind us even though we haven’t been playing that well recently," he said via the Liverpool Echo.
"They really sucked the ball down towards that end in the second half and, on any other day, it would have been an Everton win because we created the chances to do it. Our fans created a fantastic atmosphere. It honestly gives you such a lift as a player when you have more than 30,000 Toffees willing the ball down that end.
“It definitely gave us an extra bit of belief, extra yard of pace and energy - and we’re just devastated we couldn’t reward them with a fantastic day. It really hurts.”
Winning the FA Cup may also have been Martinez's last chance to save his job on Merseyside. He is unlikely to have done his chances of surviving the summer much good at all, but time will tell.