Jose Mourinho's side overcame their Sky Bet League One hosts 2-0 at the County Ground but the night was tinged with disappointment.
Van Ginkel - making only his second Blues start following a summer switch from Vitesse Arnhem - went down after an early challenge from Town midfielder Alex Pritchard and, despite briefly trying to carry on, soon hobbled off in the 10th minute.
He was replaced by fellow midfielder Ramires, who added to Fernando Torres' opener with Chelsea's second goal of the night, before then being forced off himself at half-time.
But it is Van Ginkel's injury which is providing the greater concern.
Assistant first-team coach Steve Holland said: "We've got an injury to Marco which, it's a little bit too early to be certain but it doesn't look good.
"He'll have a scan on Wednesday on an injury to his knee. It's too early to say (if it's a bad one) but let's hope that's not the case. It didn't look good as he came off so early in the game. We'll have to see the results of the scan, we'll know more on Wednesday.
"And his substitute, Ramires, also had to go off with a muscular injury at the top of his hip which was causing him more discomfort as he tried to run it off. He'll be a major doubt for Saturday at Tottenham, not out for sure yet, but a major doubt.
"They're the two disappointments of the evening."
Swindon put up brave resistance against top-flight opposition and could be proud of their efforts, with Dany N'Guessan seeing a header correctly ruled marginally offside just after the break.
The Robins now return to league action and an assault on promotion under boss Mark Cooper, who has only been in the job on a permanent basis for a month after initially arriving as assistant manager before then taking the caretaker reins over the summer.
In the build-up to the cup tie Cooper admitted he was hoping to pick the brains of respected Chelsea boss Mourinho after the clash, but he revealed he did not get the opportunity to do so.
"He just said 'well played, your team played well and best of luck for the season'," Cooper said.
"I know he's got a million people pestering him before the game, at half-time and after the game. I completely understand."