The Scottish Premier League side are back in the main stages of Europe's premier competition for the first time since the 2010-2011 season, having prevailed 3-2 on aggregate against the Dutch giants.
Van Bronckhorst's men held out for a hard-earned 1-0 win at Philips Stadion, with Antonio Colak's 60th-minute strike settling the tie and booking their place in Friday's (AEST) draw.
And the Rangers boss was delighted by the efforts of his players, and the control they demonstrated throughout the second leg.
"It feels great to win this game, to get the whistle at the end and to achieve what we wanted," he said.
"I was more relaxed than last week because of the way we played. It was very controlled, we played the way we wanted. We scored in the right moment and became stronger and stronger in the game.
"As a player and coach, you're only thinking about competing with the best teams in Europe, and we have the opportunity to do so.
"I'm really happy with the way we played. I couldn't be prouder at the moment."
Meanwhile, matchwinner Colak was reduced to tears after the final whistle. The Croatia international has now scored five goals since arriving from PAOK, although this is undeniably the most important yet.
And Gers' number nine cannot wait for his maiden appearance on Europe's biggest stage in club football, telling BT Sport: "It's difficult to describe in words.
"It was a great team performance. It's an amazing journey we'll have this year, and I'm so proud of all of these guys.
"I was in tears on the pitch, it was just a moment of joy. As a kid, I always wanted to compete at this level, and to be here is amazing."