Liverpool's first-choice left back suffered a dislocated shoulder during Scotland's defeat to Spain during the October international break.
The 29-year-old left the pitch in clear discomfort, after a collision with Unai Simon, with his arm supported by his shirt in a makeshift sling.
Fears over the severity of the injury quickly grew, while Klopp confirmed prior to Liverpool's Merseyside derby win that Robertson would undergo surgery to help prevent a reoccurrence of the injury.
There has been no official word on the timeline of Robertson's recovery, though many have predicted that the former Hull City star will be unavailable for the remainder of the calendar year while he recovers.
Speaking in his pre-match press conference ahead of Liverpool's Europa League clash with Toulouse, Klopp revealed that Robertson went under the knife on Wednesday morning – with all going to plan.
He said: "Robbo had surgery and all went well, as good as it could be.
"Rehab starts as soon as he wakes up, I think it happened this morning."
Meanwhile, regarding the Reds' other big injury doubt, Klopp confirmed that Cody Gakpo is fit again, having resumed full training, and could play a part against Toulouse.
He said: "Cody is in full training now since Sunday. He is available for the squad."
Liverpool are expected to make numerous changes to the lineup midweek, with a Premier League clash against Nottingham Forest to follow just three days later on Sunday, 29 October.
Klopp insisted, however, he will not make 'too many' changes with the squad aiming to go deep into the competition.