Sung, 29, carded seven birdies and an eagle to reach 16 under par, a new 36-hole tournament record.
First-round leader Rickie Fowler shares third, while ex-world number one and fellow American Jordan Spieth missed the cut after shooting 77 on Friday.
Englishman Justin Rose is 10 shots adrift after carding a round of 71.
Sung has missed the cut in five of his nine PGA Tour events in 2017 but holed a 30-foot eagle putt on his penultimate hole, the par-five eighth, to equal the course record at the Golf Club of Houston in Humble, Texas.
A first PGA Tour victory on Sunday would also secure the final place in next week's Masters, but he said: "My experience is if I try to do something and think about the result it doesn't work out very well, so I'm taking it one shot at a time and the results will come."
The cut came at level par, one too many for former Masters champion Adam Scott of Australia - who had six bogeys and a double bogey in a 77 - and Englishman Andrew "Beef" Johnston, who mixed four birdies with four dropped shots in his 72.
Spieth also had six bogeys in his 77, while Open champion Henrik Stenson of Sweden finished five over par after two double bogeys in a 75.
Englishman Lee Westwood double bogeyed the last to close on seven over.
World number 202 Sung Kang of Korea produced a superb round of 63 to take a commanding six-stroke lead after two rounds of the Houston Open.