Spain went into the contest knowing a victory would have secured their passage into the competition's final four, but they were in trouble at half-time at the Benito Villamarin in Seville.
Sterling scored twice to sandwich a Rashford strike, but Kane played a role in all three with some fine play, leading from the front as captain.
Spain responded after the break and did pull two goals back through Paco Alcacer and Sergio Ramos – who found the net with header deep into addition – but England's excellent first half gave them too much to do and Southgate was ecstatic.
"They should be incredibly proud of their performance," the England boss told Sky Sports.
"We spoke to the forward players in particular before the game to remind them what a threat they are, and the three of them were outstanding, causing all sorts of problems all night, a tremendous threat.
"Then it was up to the rest of the team to come out and play, be brave, and we did that really well.
"[The 4-3-3 formation] allowed us to press better, but you still need the players to play, receive the ball under pressure.
"If you can get out of their initial press quickly, then we are a threat on the counter. The finishes were excellent. I'm delighted and now we have to keep building."
Behind Monday's win, England now sits two points behind Spain in the Nations League with both teams still having to face Croatia to close the group stage.