Rashford hit a stupendous free-kick and ended a mazy run by finishing through back-up goalkeeper Danny Ward's legs after Phil Foden had doubled England's lead in what proved to be a 3-0 stroll against their British rivals in Al Rayyan.
The forward told BBC One: "Moments like this, this is what I play football for. The biggest moments, the best moments.
"I'm really happy today, I'm happy that we're going through to the next round.
"Hopefully we can build on this performance because I have massive ambitions for this team and I think we can play even better than we showed today.
"It's obviously a great feeling.
"We were a little bit disappointed as a team after the last game against the USA (Friday's goalless draw).
"I thought we could have played a lot better and the only way to bounce back from that is to have a good performance in your next game and I think we did that.
"First half, we defended brilliantly, we didn't really give them any opportunities.
"And it was just about us killing the game off and taking the chances when they came."
On his free-kick opener, Rashford added: "I fancied one in the first half but the one in the second half was in a better position.
"And from there, it's just about being calm and trying to execute what you practice in training.
"They don't go in often but I'm pleased that it went in today."
Speaking about Rashford, Southgate told BBC One: "It's great for him. He's trained really well, I have to say.
"He's really been impressive since he came back in with us. He could have had a hat-trick really, the chance in the first half, the one at the near post in the second.
"The free-kick, that is what he's capable of, it's an incredible strike really."