The Ibrox side suffered their first defeat at McDiarmid Park since March 2010 after goals from James Brown, the first of his career, and Nicky Clark gave St Johnstone a 2-0 lead. Skipper James Tavenier cut the deficit for the visitors but they left empty-handed.
Around 50 Rangers fans gathered at the front of the stadium to vent their displeasure afterwards.
The Ibrox side have now dropped five points in their last three league games while their Champions League campaign was officially confirmed as the worst of all-time after the 3-1 defeat to Ajax last week.
Van Bronckhorst knows he is in a fight to save his job. “If you lose 2-1 against St Johnstone away, for me there is no excuse,” he said.
As for the pressure now bearing down on him, he stressed it was already there “after the last couple of weeks”.
He added: "Of course, a defeat doesn’t help. But we have to continue. It is a tough one to take."
The manager insisted that he was more determined than ever to turn things round at the club. Rangers host Hearts at Ibrox on Wednesday and then face a trip to St Mirren next weekend. There are already growing calls from supporters for Van Bronckhorst to be removed.
“The determination is always there,” he said. “If you work hard, that is all you can do. It doesn’t mean you always win matches. You have to prepare the team well and today, especially the first half hour, we did really well. In the end we played more against ourselves than the opponent.
“Losing three points is always damaging, no matter which stage of the season,” he added. “Of course, if you see our last three games we have dropped five points, that’s too many. It’s not acceptable for a club like Rangers.”
On whether Rangers can afford any further slip-ups if they wish to overcome leaders Celtic, he said: “We always need to be perfect”.