The Blades went into Monday's match with a one-goal advantage from the first leg but lost 2-0 at Huish Park, where Kevin Dawson and Ed Upson scored the all-important goals for Yeovil.
Dawson opened the scoring inside the opening 10 minutes while Upson headed the decisive second on 86.
Jamie Murphy had earlier come close to snatching a goal for the visitors only to rattle the crossbar.
Morgan admitted: "It's a cruel game but football has the knack of kicking you right where it hurts and when you least expect it.
"But I can't criticise the players, they have hearts the size of lions and will learn from this."
And Morgan does not know if he will remain the manager at Bramall Lane, having replaced Danny Wilson who was sacked in early April.
He added: "There are obviously big decisions to be made.
"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."