Barrow AFC say a supporter was arrested after a Sutton player was the victim of racist abuse at Holker Street.
The Bluebirds issued a statement after their 1-0 victory.
They said police were quick to act upon an allegation during the game.
Barrow said: "The club can confirm that during Saturday’s game against Sutton United, an allegation of racial abuse was made regarding comments aimed by a spectator towards a visiting player.
"We would like to place on record our gratitude to Cumbria Constabulary for their swift actions in the wake of the allegation, during which they were able to make an arrest.
"The club want to make it absolutely clear once again that we utterly condemn discrimination in all its forms and will seek the maximum penalty for anyone found guilty of such conduct.
"We will continue to co-operate with all the relevant authorities where required."
Barrow won the game 1-0 - a result which, coupled with Oldham's defeat to Salford, sealed the Bluebirds' League Two safety.
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