Robinson scored twice in the game at Stamford Bridge, playing an integral role in the win - and subsequently received a number of Instagram posts including racist abuse toward the forward.
The 26-year-old shared images of the messages on his own account following the game.
In a statement issued on the West Bromwich Albion website, the club said they were 'appalled' by the messages, and contacted West Midlands Police.
"West Bromwich Albion has contacted West Midlands Police to report racist social media posts directed at Callum Robinson following Saturday's 5-2 win over Chelsea." the statement read.
"The despicable Instagram posts were sent in reply to the Republic of Ireland international after he scored twice in a memorable team performance at Stamford Bridge."
"The club is appalled by the messages and will offer Callum the help and support he may require in response to the incident.
"Albion will continue to confront all forms of discrimination and will assist the authorities with their inquiries, seeking the toughest available legal punishment.
"It is deeply disturbing that scoring two goals in a football match can provoke a reaction of this kind."
Robinson is the latest footballer to be targeted online, with social media companies receiving heavy criticism over their response to such issues.
Marcus Rashford, Anthony Martial and former Aston Villa loanee Axel Tuanzebe are just a few other high profile footballers previously targeted on social media.
Recently, Walsall winger Tyreik Wright, on-loan from Villa, has also suffered from racial abuse - with evidence handed to the Crime Prosecution Service as part of the investigation.