The Killie boss had been on the receiving end of racist abuse via a letter sent to the club on Monday morning.
The club released a statement condemning the “disgusting act”, while noting they were working with Police Scotland to identify any individuals.
Kilmarnock have had a tough time on the park of late, losing eight of their last nine games prior to the 2-0 win at Fir Park.
After the match Dyer told Sky Sports: “What’s happened to me, it’s not about Kilmarnock. The Kilmarnock people have been very good to me. It’s about someone who has got a grievance against my colour, it’s as simple as that. It's nothing to do with Kilmarnock. It's nothing to do with managing a football club.
"If they turned to me and said ‘I don’t want you at this football club because you are not doing a good job’, I’d take that and get on with it. But this is someone who’s got the front to write a letter because of the colour of my skin, as simple as that.
"It's important we carry on doing what we do. We take the knee, we keep getting the message out there and hopefully one day there will be no more.”
He added: “I am in a privileged position, I know that. I can bat it off, it’s not hard for me. I can talk about it and people will geta round me. But there are people out there who don’t have nobody so it’s for the likes of me to keep talking out, speaking about it and make sure we do the right thing.”
Dyer’s interview was praised by Beale who expressed his pride and delight for the Killie boss on Twitter.
He posted: “A very good interview from Alex Dyer who shows his class as a man and speaks extremely well about a very difficult week for him personally. He is someone I have known for a long time and I’m delighted his team got a positive result for him today.”