Strikes from Jonas Hofmann and Marco Reus ensured that a late Robert Lewandowski own-goal did not deny die Schwarzgelben all three points in their first home game of the Bundesliga campaign.
Klopp admitted he had been fearful about how fatigued his players would be after the midweek international games but, though the result was what he had expected, the BVB head coach was thrilled with the team's response.
"We absolutely deserved to win but then that's what we expected. I'm happy with the way my team played," the 46-year-old told reporters after the win.
"I didn't expect them, after the break with the national team where we were spread out all around the world, to play and bring the stars down from the sky.
"We had anticipated hard work and that's what we did so I'm happy with that."
Klopp was also impressed with Hofmann's impact after the 21-year-old scored the opener and was brought down for Reus to score from a penalty.
"To be able to put Hoffi on as a substitute with his freshness and him playing with little pressure, knowing how to use his ability and spread happiness, well, that's a beautiful story," he continued.
For Braunschweig, it was a case of fine margins in defeat again following their 1-0 loss to Werder Bremen in the season opener and Klopp praised Torsten Lieberknecht's side's work-rate and attitude.
"If a team is well organised I find that normal but if a team is very well organised I find that cool and that was the case today," he added.
"What I found remarkable is that Braunschweig kept this up for 90 minutes. I've rarely seen that that, a team that stays so organised so ready for the tight situations."