"I got off to a pretty good start, hit some really close iron shots and I made some good 10 or 15-foot putts for birdie," Curran told reporters.
That left the 28-year-old at nine-under 130, where he was joined later in the day by fellow American Jimmy Walker (66) and overnight leader Steven Bowditch of Australia (68).
Walker and Bowditch did well to finish in fading light after the start to the second round had been delayed by three hours because of water-logged conditions caused by 4.5 inches of rainfall overnight.
"Didn't want to have to get up early (on Saturday)... it will be nice to sleep in little bit," said Walker.
Thirty-three players will have to complete the second round on Saturday morning, though further thunderstorms are expected overnight on Friday.
American Ryan Palmer (66) and Australian Cameron Percy (64) were at eight under with local favourite and world number two Spieth a further two shots back after carding a 64.
"I had a lot of short birdie putts today," said Spieth after piling up seven birdies, three of them on his last three holes, and two bogeys. "I zeroed in on the targets nicely.
"The finish was certainly nice. For the middle of that round, I was only hanging around one or two under and to be able to close at five under on the day puts me in contention."
Grounds crews worked hard in the morning to prepare the course for play, though the par-four 14th had to be shortened to a par-three measuring just 104 yards because of an unplayable fairway.
Officials said they would continue to evaluate the situation and determine whether to play the 14th hole as a par-three for the final two rounds.
"They made a great decision in going out and playing that as a par-three," said Curran, who earned his 2014-15 PGA Tour card by finishing 12th on the Web.com Tour's money list last season.
"We have had such bizarre weather out here ... it was either that or not play the round."
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Peter Rutherford)
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