Van Bronckhorst’s side followed up their knockout round play-off triumph over Borussia Dortmund with a breathless 3-0 win over Red Star Belgrade in the first leg of their last 16 tie at Ibrox on Thursday night.
It leaves Rangers on the verge of reaching the last eight of a European competition for the first time in 14 years and has shortened the odds on them going all the way to the final in Seville in May.
Van Bronckhorst feels the tournament is now wide open but remains cautious about next Thursday’s return fixture in Belgrade as he warns that his players cannot afford to drop the high standards they have set.
"Any team left in this tournament can win it in the end,” he said. "But the winners will be the team that has played all the games on a high level and don't make mistakes.
"We are at a stage where there are very strong opponents left in the tournament. The final is still far away - there are a lot of games to be played.
"For me, the most important thing in Europe is that with every game you play - and every tie - you have to be there. One slip up, one night when you aren't there, can cost you your next round.
"The most important thing is Belgrade next week. If we reach our levels and performances, we have a big chance to go to the next round.”