England edged ever closer to reaching the 2015 European Championships with the win, which put them nine points clear of second-placed Moldova in the group stages.
Southgate singled out Nathan Redmond for special praise after the Norwich City man netted a hat-trick, but he was also keen to pay tribute to his whole team.
"The one area of [Redmond's] game he is looking to add is goals, and I think we are able to give him the freedom to play in more advanced areas. I thought he was such a handful all night," Southgate told Sky Sports News.
"We have got three or four players that if we can get in one-versus-one situations, then they are devastating in attack. Tonight, his finishing matched the rest of his game. I thought we were terrific with the ball, the way we moved it, the pace. We created so many really good chances.
"Probably, if you were to be critical, we should have been more ruthless, but they are a fantastic group of players to work with. Everything you ask them do do, they take on board. I thought there were some outstanding performances. We want to make sure we keep putting markers down, that we are a side that teams will fear."
Redmond also scored the only goal of the game in the reverse fixture between England and Wales three months ago.