The Cottagers opened up a 2-0 lead in front of their own fans, but Derby staged a second-half comeback to book their place in the fifth round.
Symons believes that the turning point came in first-half stoppage time when a Chris Martin penalty saw the visitors get back in the match.
"In the first half we were well and truly in it, but the goal on the stroke of half time gave Derby a huge lift and top teams don't need lifts like that," he told BBC Sport. "We nullified them in the first half but the second half was a different story and Derby punished us."
Fulham continue their Championship campaign on Saturday when they make the trip to Wigan Athletic.