The Scottish champions were unable to build on an early opener from Manchester United loanee Amad Diallo on his debut as mistakes from goalkeeper Allan McGregor and centre half Calvin Bassey were punished by Jordan White and Regan Charles-Cook to give the hosts a half-time lead.
Rangers hit back strongly in the second half and looked to have salvaged the situation when goals from James Tavernier and Connor Goldson put them 3-2 up.
But their failure to defend a 96th minute corner kick, with McGregor again looking suspect, allowed County substitute Matthew Wright to bundle in the home side’s leveller.
Van Bronckhorst made no attempt to hide his dismay at the standard of his team’s display.
“You have to be decisive in the moment,” he said. “You can talk afterwards, but you don't need to make the mistakes.
“If you go 1-0 up then you have to be solid and make sure you do not give any chances away. Instead of doing that, we made two mistakes and let them go 2-1 up.
“The reaction in the second half was good but in the end we gave two points back when we should have had three points.
“At the end, in extra time, we were too sloppy. We give the ball away, we give a corner away and concede late in the game.
“For us, that is not something I am used to seeing - so many goals against us in this manner.
“Today many things went wrong, especially defensively and it was a disappointing afternoon for us.
It was a hard game but we started really well and got the first goal after a couple of minutes. Then we gift two goals to Ross County and that made it difficult for us.
“In the second half we started really well and dominated the whole half. We created a lot of chances and went in front again at 3-2. We had some chances to kill the game and get the fourth and decisive goal.”