Recent signing Van Ginkel suffered a potentially serious knee problem early in the 2-0 third-round win and was replaced by Ramires, who failed to appear for the second half because of a hip injury.
The Brazilian could now miss Saturday's derby game at Tottenham while, according to Jose Mourinho's assistant Steve Holland, the prognosis for Van Ginkel could be worse.
Holland said: "It wasn't perfect because we've got an injury to Marco. It's a little bit too early to be certain but it doesn't look good.
"He'll have a scan on Wednesday on the injury to his knee, and his substitute Ramires had to go off with a muscular injury at the top of the hip in the gluteal area which was causing more discomfort as he tried to run it off.
"He'll be a major doubt for the Tottenham game at the weekend. They're the two major disappointments of the evening."
Chelsea right-back Cesar Azpilicueta later tweeted his concern for Van Ginkel, who appeared to twist his knee in the first-half incident at the County Ground.
He wrote: "Happy with the victory but sad for Marco's injury. Get well soon my friend. We are all with you."