Ed Upson's decisive headed six minutes from time added to Kevin Dawson's early opener as Yeovil overturned a 1-0 first-leg deficit at Huish Park.
Morgan's side were inches away from celebrating an equaliser when Jamie Murphy's curling effort rattled the crossbar.
But despite introducing Dave Kitson to boost their attacking threat the Blades struggled to test Glovers goalkeeper Marek Stech as time ran down
And Morgan admitted the timing of Yeovil's winner was tough to take.
"It's a cruel game but football has the knack of kicking you right where it hurts when you least expect it," he said.
"But I can't criticise the players, they have hearts the size of lions and will learn from this."
Talk soon turned to Morgan's future having taken charge of the Blades since Danny Wilson's departure last month.
"There are obviously big decisions to be made," he added. "I feel over the three weeks that I've had the job that I feel we've done well here. But that isn't my decision. They know I want it."