Cole said sorry for his studs-up high challenge on the Spurs player after watching the incident on television the morning after Wednesday night's thrilling 4-4 draw at White Hart Lane.
Cole, who was also criticised for turning his back on referee Mike Riley as he was shown the yellow card, included the official in his apology.
"First you need to give credit to Ashley because he behaved like a mature guy," said Grant.
"He saw what he did the day afterwards and he apologised himself. Nobody pushed him to apologise and you need to give credit to him about that.
"He made a tackle and he realised he needed to apologise about that and his behaviour. It is the first time he's done that kind of thing and he apologised.
"In football, unfortunately things happen, there are worse tackles than this.
"It's more important not to do it, I agree, but because it's football and it happens, it's more important to take responsibility.
"You need to give credit to a player who after only 24 hours saw it on TV and took responsibility for it."