Simon Silwood, 50, targeted Sawyers, who moved to Stoke City in the summer, after West Brom lost 5-0 to Manchester City in a Premier League match in January. Birmingham Magistrates Court found him guilty of having posted a "grossly offensive message."
West Brom banned Silwood for life earlier this month.
The Crown Prosecution Service (CPS) said that during his police interview, Silwood admitted to posting a message on social media after becoming upset by the 5-0 defeat but blamed an error made by using predictive text.
The judge said his message was "motivated by or demonstrated hostility based on race."
"Silwood deliberately chose a word known to have clear racist connotations to abuse a footballer due to the colour of his skin. Such behaviour will not be tolerated in football and will be prosecuted by the CPS," Mark Johnson of the CPS said.
The court ordered Silwood to pay £500 ($684) in compensation and £500 in court costs.
West Bromwich were relegated at the end of last season.
Several players have been subjected to racial abuse online in recent months, especially in the Premier League, and English football has been piling pressure on social media firms to do more to tackle the problem.