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.
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.
"What was really pleasing was that when we went behind we stayed calm and used the ball calmly from the back. In other big games, we have lost the ball too easily.
"Our forwards worked so hard. When your front players play like that it is a great thing for the team.
"There were times in the first half we were passive near our own goal. The biggest difference in the last couple of years is that we have more legs and physicality in midfield. You need to be outstanding without the ball against world-class teams."
The England boss was quick to revert back to his usual grounded self though and was keen to urge the importance of maintaining his side's winning run.
"We need to make sure we are one of the top seeds next month, we need to win our next two games. We want to be in control of that," he added.
"We need to keep building. There is more to come from this team. This was the toughest qualifying group, with the seedings. People have criticised us for not beating the top teams enough but we have performed really well."
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."
Four-time world champions Italy host bogey team North Macedonia in their game in hand on Ukraine before taking on the war-hit nation in Leverkusen in a crucial final match.
England started brightly on Tuesday but were caught out in the 15th minute by a sweeping Italian move which ended with Scamacca tapping home his first international goal from Giovanni Di Lorenzo's low cross.
But the home side levelled shortly after the half-hour mark as Kane planted a penalty into the corner after Bellingham had been fouled in the box.
England only needed a point to qualify but completed the turnaround 12 minutes after the restart as the rampaging Bellingham won the ball in midfield before feeding Rashford, who cut inside and lashed a shot past Gianluigi Donnarumma.
Kane put the result beyond any doubt in the 77th minute, powering through the Italy defence and slotting home his 61st England goal.
