The Tottenham forward had not scored before at a major tournament after drawing a blank at Euro 2016, but Southgate - who joked about the striker's well-documented August struggles in front of goal - was full of praise for the Three Lions captain.
"If he hadn't scored, I'd be answering questions about his ability to score in tournament football," he said.
"The only thing he hasn't done now is score in August. He's moved every other barrier, and I'm personally delighted for him.
"I know he will feel the pride in leading his country to a win in a World Cup match, which is the most important thing."
Following that game, Southgate's men will face Belgium, who defeated Panama 3-0 in their opening game on Tuesday.