Charlie Austin looked to have salvaged a point for QPR as he cancelled out Ryan Bertrand's 54th-minute opener with a fine turn and volley.
However, a magnificent overhead kick from Italian striker Graziano Pelle settled the game in Southampton's favour, making it six successive wins for Koeman's side in all competitions.
Saturday's Premier League triumph came on the back of a terrific 2-1 League Cup success at Arsenal and Koeman was delighted with the performance of his players.
"We knew before the game it would be tough," Koeman said. "All the credit to QPR because they played a fantastic second half.
"They played with a lot of forwards and long balls and it was difficult.
"If you show that character after the win of Tuesday, for me as a manager that means a lot about the ambition and the spirit of the squad."
The Dutchman, who succeeded Mauricio Pochettino at St Mary's Stadium in June, also lavished praise on Pelle for his overall form and a goal that will surely rank among the best of the campaign.
"It was a great finish," Koeman added. "It will be one of the best goals of the season.
"I'm happy about his performance from the start of the season because he shows his qualities and his personality.
"He's always keeping the ball for the team and we can push up with him. That's important for the team and how we play. If you score like this, you can't do it better."
