Italy had taken the lead against the run of play in the 15th minute through former West Ham forward Gianluca Scamacca, but the Three Lions showed resilience to come from behind in front of an at times subdued Wembley.
Two goals from Harry Kane and one from Marcus Rashford - as well as an impressive all-round display by Jude Bellingham - saw England run out as deserving winners against Italy.
With two group games still to play, unbeaten England, winners of five of their six qualifiers, are guaranteed to finish top of Group C.
Speaking to Channel 4 after the match, Southgate was full of praise for his players: "They're incredible to work with, I have to say.
"So hungry, so eager to learn, so together, you saw that in the performance tonight.
"We talked before the game about the mentality being the biggest thing in these matches, maybe you've got to come through setbacks, whatever, you've got to be relentless. I thought the performance was relentless tonight."
England captain Kane told Channel 4 he was pleased with the calmness shown in the display: "Qualifying is never easy. Sometimes we are just expected to go to the big tournaments but we have seen in the past it is not as easy as it looks.
"We went 1-0 behind and stayed calm. We knew we could get the winner and we have done it.
"To qualify with two games to go, credit to everyone involved."
He also feels Wembley has become a difficult place for any side to come: "It is our home, we have made it a real fortress over the years.
"We want to repay our fans with victories and we have done that today."
