Van Ginkel - making only his second start following a summer switch from Vitesse Arnhem - went down after an early challenge from Alex Pritchard and, despite briefly trying to carry on, soon hobbled off in the 10th minute.
It understood reports in the Dutch media that the 20-year-old will have surgery on Thursday and is expected to be out for six to nine months are accurate.
Chelsea paid around £8million of Van Ginkel in July, but it looks as though his season has now ended after seeing just 97 minutes of action.
The Dutchman showed glimpses of his potential during his only other start against Basle in the Champions League, and he also looked impressive when coming off the bench against Hull and Aston Villa in the Barclays Premier League.
Chelsea assistant first-team coach Steve Holland said he feared the worst for Van Ginkel after the Capital One Cup clash at the County Ground, which Chelsea won 2-0.
"It doesn't look good," he said.
"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."
The injury also severely hampers Van Ginkel's chances of making Holland's World Cup squad.
The 20-year-old made his international debut last November and was expected to challenge for a place in Louis van Gaal's squad for Brazil.
Chelsea later confirmed the Holland midfielder had undergone an operation.
A statement on chelseafc.com read: "Chelsea Football Club can confirm Marco van Ginkel successfully underwent anterior cruciate ligament surgery today (Thursday).
"The 20-year-old midfielder is expected to be out for approximately six months.
"Van Ginkel suffered the knee injury during the first half of our Capital One Cup match at Swindon on Tuesday evening."