Pickett pounced on the crumbs of a marking contest in the dying seconds of Saturday night's clash at the MCG and floated a snapshot through the goals under pressure.
It sparked wild scenes for the Demons, who took a step closer to locking in a top-four spot with the thrilling five-point win.
But there was a sour aftermath when Indigenous speedster Pickett received abusive messages via social media.
It is the second known time Pickett has been the target of racial abuse in the past two seasons.
"Last night a Melbourne player was again subjected to racism on social media," Melbourne said in a statement on Sunday.
"This sort of behaviour is abhorrent and needs to stop.
"It is saddening, angering and unacceptable that this behaviour continues to occur.
"It is important that we, as a football community, call this behaviour out and hold these individuals to account.
"It is simply not OK to remain silent."
Melbourne are working with the AFL integrity department to see if the social media user responsible for the comments can be identified.
The racial abuse levelled at Pickett came just days after Indigenous Brisbane utility Calllum Ah Chee was targeted in a similar manner amid the fallout from his on-field collision with Carlton captain Patrick Cripps.
Carlton's Muslim defender Adam Saad was also targeted last month by a spectator at Adelaide Oval.
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