The young Blues fell behind to Chuba Akpom's 11th-minute opener at Underhill, but responded with aplomb to record a 4-2 victory to reach the quarter-finals.
Matthew Pennington levelled the scoring moments later, before Chris Long, Conor Grant and Harry Charsley put the Gunners to the sword during a whirlwind start to the second half.
And speaking to evertonfc.com after the contest, a thrilled Sheedy was full of praise for his united squad.
"It was a magnificent performance," said the Blues legend.
"The players are getting better and I told them before the game that they will continue to get better. The determination, the camaraderie and team spirit that we've got shows in abundance on the pitch.
"There's a good unity. It makes that league defeat to Fulham a bit more galling because we asked our players to play 36 hours after a massive effort against Port Vale in the last round and we lost our unbeaten run, which stretched back a long way.
"That still hurts me, but for the game tonight, Arsenal put their best team out and we've totally dominated the game. There were a lot of great characters out there. I'm absolutely delighted for them. They're buzzing!"