The 104th meeting of the old rivals ended with last year's European Championship runners-up wrapping up top spot in Group B on Tuesday evening at the Ahmad Bin Ali Stadium.
Rashford's brilliant free-kick and fizzing strike came either side of Phil Foden's goal in Tuesday's 3-0 triumph against Wales, who head home as England gear up for the last 16.
Their reward for topping the group is a clash with Senegal on Sunday and alcohol-free celebrations in Qatar, having toasted progress with a beer in Russia four-and-a-half years ago.
"We've got loads of milkshake back at the hotel," Southgate said with a smile.
"Everything is different, isn't it? We're embracing it. We love our base camp, the people that are operating the hotel are fabulous people. Really we couldn't be happier.
"We're enjoying each other's company and the players are really tight, so to this point we're very pleased.
"Of course the big business starts now, really."
Foden did his chances of a greater role on Sunday no harm in Al Rayyan, but it was Rashford who made the greatest case to start against Senegal.
The 25-year-old has three goals in as many games at this World Cup, with his goal off the bench against Iran coming on his first England appearance since missing his penalty in the Euro 2020 final shootout.
Rashford struggled for form and fitness in the intervening period, but the Manchester United forward has kicked on impressively.
"It's been a challenge for him," said Southgate, who was unaware that the man-of-the-match's friend Garfield Hayward, to whom Rashford dedicated his goals, had died in the build-up to the game.
"I went and saw him in the summer, had a long chat with him and he had some clear ideas on things that he felt he needed to think about and to do.
"You can see with his club there's been happiness in his performances this year.
"And I have to say that's shown itself on the training ground all the time with us.
"We've got a different version completely to the player we had in the Euros last summer, so he's managed to produce those moments tonight.
"He could have had a hat-trick, really, with the chance in the first half and then the one at the near post towards the end.
"But great for him and because it's great for him it's great for us."
As England begin preparations for the Senegal clash, Wales pack their bags for home.
Robert Page's side finished bottom of Group B with one point from three games - a 1-1 draw with the United States preceding a 2-0 defeat to Iran.
Wales captain Gareth Bale, who pushed away a cameraman at the final whistle who got too close to him, told the BBC: "Everyone's disappointed but we're all proud of what we've achieved to get here.
"We have to look how far we've come. Of course we wanted to get out of the group, but football's difficult.
"We didn't live up to our expectations at this tournament.
"If you'd told us two years ago that we would play in a World Cup, we'd have all pinched ourselves.
"We'll all walk out of that changing room with our held high and proud of each other.
"It's been difficult at this tournament, we haven't played very well. But we've given everything and we won't have any regrets."
Bale had a disappointing World Cup and did not reappear for the second half against England, with boss Page saying he had been "struggling with his hamstring".
But the skipper, who had little game time for his American club Los Angeles FC ahead of the World Cup, stressed that he has no thoughts about ending his Wales career.
"I'll keep going as long as I can and as long as I'm wanted," he said. "It's a difficult moment now but we go again in March (in the Euro 2024 qualifiers)."