Dyer was desperate to take the kick to complete a unique Wembley hat-trick - the first-ever in a League Cup final.
But club skipper Monk said: "Nathan is going to get a lot of stick this week.
"I need to see it again as it was a blur but he'll be getting hammered, don't you worry.
"Nathan will handle it though because he's class. He and Jonathan are good mates. I'm so happy for him that he played such a big part in our win.
"It's been an indifferent season for him really and he's not been a regular like he was in the past. But he's a good lad, and he played great."
Defender Ashley Williams had the first tongue-in-cheek dig at Dyer. He said: "He's all right now but we know what he's like, he's a little sulker.
"He wanted his hat-trick, but Jonno is greedy too! No one knew who the taker was. It's been sorted out and it was no big deal. They are good mates."