With scores locked at 1-1 after 90 minutes, Van Bronckhorst brought the two on in the 117th minute with penalties in mind. Ramsey was the only player to not convert his penalty, Kevin Trapp saving with his feet at 3-3, before Rafael Borre followed Roofe's spot-kick to secure a 5-4 shootout win for Eintracht.
It was the Gers' second Europa League final loss in as many appearances, following 2008's 2-0 loss to Zenit in Manchester.
According to Van Bronckhorst, despite evident disappointment in the changing rooms afterwards, his final list of penalty takers was impacted by a combination of factors including Borna Barisic's inability to take one of the penalties, and pre-match preparation.
"It was tough, physically, but the players gave everything and gave all their effort on the pitch," Van Bronckhorst said post-match. "I subbed some players because they were struggling physically, but I cannot complain. They gave everything and that's all you can ask as a coach.
"Borna had to go out and he's one of the first penalty kickers in the game. We trained in penalty kicks, because some players are comfortable taking them. We had a good feel of players who wanted to take a penalty and players who didn't. In the end we had our list and we had to adjust it because of the subs.
"You could see it after the game straight away, it's never a nice feeling. Everyone is very disappointed and you can sense that in the locker room. I think it's normal, so soon after the game, but Aaron took responsibility to take the penalty. Unfortunately he didn't make it, but you want players who are comfortable and who are ready to take them."
Rangers created sporadic opportunities, with Joe Aribo's 57th minute opener at the top of the penalty area one of their only four shots in the box.
The game petered after Borre's equaliser in the 69th, but Ryan Kent had the best chance to win the match in extra-time, only to be denied by Trapp from close range under pressure from Kristijan Jakic.
Ultimately, Van Bronckhorst asserted he could not fault the effort of his players, and understands their post-match predicament more than most.
"Especially in those minutes near the end of the game, it's decisive," he said. "It's a big chance for us, but Ryan did everything he can to score the goal. In the end, you know when you have chances you have to take them."
"If you play a final in Europe and you lose, it's going to hurt, because if you play a final you will do everything you can to win it. In the end, with penalties, it's a lottery and tonight we weren't on the good side.
"But I can't complain with everything my players gave and in the games before tonight. In the end, I think it was a really tight game. Went all the way to penalties and, you know, we lost. A big disappointment because we were so close to winning a trophy. I lost a World Cup final, the biggest game there is, also a huge disappointment but you have to move on."