The Mexican golfer had a five-birdie stretch mid-round, then added two late birdies to finish with a 13-under total of 203 and a five-shot lead going into Sunday's final round at The Lakes.
Japan amateur Keita Nakajima was in second place after 70, while Bradley was tied for third with Australian veteran Marcus Fraser. They were six strokes behind after 71s.
Kuchar, who trailed by a stroke after the second round, shot 73 Saturday and was tied for fifth, seven strokes behind Ancer.
Ancer, making his first trip to Australia, said the conditions were difficult.
''The winds were blowing all day so I had to stay patient and keep the ball in play,'' Ancer said.
''It was a very challenging round but everything went my way and, whenever something didn't go my way, I managed to make at most a bogey which, out here, was pretty good.''
Bradley said he hopes for a comeback on Sunday.
''I wish I wasn't that far back obviously, but you never know,'' Bradley said. ''This is a tricky little course. You can go out and shoot a low one out here, so I look forward to doing that tomorrow.''
Kuchar agreed the conditions were tough.
''I had a couple disappointing holes, but that's going to happen in conditions like these,'' he said. ''What Abraham did today is just remarkable, I did not see that at all today. That's just fantastic golf. He's got a big lead and deservedly so.''
Ancer said he'll try not to think about that big lead on Sunday.
''Obviously it's always nice to have a cushion, but you don't want to be thinking about that too much,'' he said. ''You just want to go about your business and keep the same game plan for tomorrow that I've had every single day.''