The Welsh continued their unbeaten start in Group B on Monday night as they earned a 2-1 victory in Cardiff.
Despite the nation failing to reach a major tournament since 1976, Coleman believes that the spirit within the current squad could see them through this time around.
"The group has really matured. A lot of those that have been around a while have really matured and the younger ones have got belief," he told Sky Sports News. "The togetherness is incredible.
"We're greedy now. We want the points and we want to get them no matter what we have to do to get them, and we've done that. It was a great three points."
Wales continue their qualifying campaign on November 16 when they face Belgium in Brussels.