Derbyshire closed Day 3 at Swansea on 504/4 as Billy Godleman finished the day on 211* and Harvey Hosein also remained unbeaten on 53.
Derbyshire had resumed play on 221/2 but the breakthrough finally came on 365 when Tom Lace fell for 143 thanks to a superb catch in the deep by Morgan.
"I can't fault the effort from the lads, they tried all the way through and had loads of energy in the field," said head coach Matt Maynard at close of play. "If we're honest we weren't at our best with the ball so far. There were a few too many bad ones but every now and again that happens. The bowlers have bowled outstandingly all year."
"Lukas Carey bowled really well but we put down a couple of chances in the field. Godleman's made the most of the three chances he's had and he's there at the end of the day. Whatever happens tomorrow we know we have to fight and when we get out there we need to stand up as a unit."
"It's a belter of a pitch, the heavy roller flattened it out and while there is still a bit in it, the margin for error is so small."
"We bowled well with the 2nd new ball but we fell away from that. I can't fault the lads efforts and sometimes you try too hard, and that's happened with Michael Hogan in this game. It's quite unusual for him (his figures), he's just tried to take wickets rather than do what he did last week (against Northants) which was hold his length superbly.
"Everyone can have a bad game, we've been really good all year and as a batting group we have a job to do tomorrow to save the game."