Van Garderen attacked a group of overall contenders - minus overnight leader Wilco Kelderman - with around 4.6km to go on the hors category Rettenbachferner climb above Solden.
The American was able to hold off Miguel Lopez (Astana) and Warren Barguil (Giant-Alpecin) for the win.
"It was a hard moment and I saw that some of the other riders didn't look so strong and I was feeling good so I thought it was a good opportunity to attack and go for the stage victory," Van Garderen said.
"It's definitely up there with the hardest climbs I have ever done. (But) the feelings were good. They have been good all race except for yesterday when I had a bad moment and got really cold. It's still very regrettable that it happened because I feel I have the form to win this race overall."
An early breakaway of three riders stayed away until the base of the final climb where the peloton faced a 11.5km climb with an average gradient of 10 per cent.
Now the race leader, Barguil was happy with how he handled the stage.
"In the finale, I just followed the main group and when Van Garderen attacked I was expecting Team Sky to accelerate, but they didn’t.
"So I decided to go by myself to try and close the gap. Unfortunately, I couldn’t and then I just gave everything I had to reach the finish line.
“At the start of the final climb, I still had strong support from the team and we managed to set a good tempo in order to decimate the bunch as much as possible. They did a great job."
Van Garderen believes that with just two stages remaining, including a 17km ITT tomorrow, he can move up the overall classification from seventh overall.
"Looking at the GC as it stands after today I definitely think I can move up a couple of places and maybe get into the top five. If I have a really good day and some of my rivals suffer a bit then the podium could be possible."