Taking to Instagram, the Italian revealed: "Surgery ok. I have to thanks all the people worried for me. Nothing else, just a big thanks. Hugs!"
Cigarini suffered a double leg fracture in the incident during Kimi Raikkonen's pit stop for which the Italian team was subsequently fined 50,000 euros.
The Italian was in the process of removing the Finn's left-rear wheel when the all-clear was given for the driver to pull away, the car running over the hapless Italian's left leg in the process breaking the shinbone and fibula.
Following treatment at the circuit Cigarini was subsequently transferred to Bahrain's BDF Hospital, 20 minutes form the Sakhir circuit, where he underwent surgery late last night.
Raikkonen, a newcomer to social media, posted the message "get well soon mate", to his crew member's Instagram account.
Ferrari is currently investigating the incident, having suffered another unsafe release with the Finn's car on Friday.
Though Raikkonen retired in the aftermath of the incident, teammate Sebastian Vettel went on to win the race.