Overnight leader Spieth, the 2015 Masters and US Open champion, carded a four-under-par 66 to move to 12 under.
Fellow American Boo Weekley, ranked 390 in the world, hit five birdies on the back nine in a round of 65 to storm into contention, before Spieth birdied the last to hold his lead.
England's Paul Casey carded a four-under-par 66 and is three shots behind.
Former world number one Spieth shot consecutive bogeys on holes 13 and 14 before birdies on the 15th, 16th and crucially the 18th handed the 23-year-old the narrowest of leads.
But while Spieth endured a mixed back nine, 2008 Ryder Cup winner Weekley birdied the 11th, 13th, 15th, 17th and 18th to momentarily take the clubhouse lead.
Ireland's Padraig Harrington is now five shots behind Spieth after an up-and-down round in which he shot five bogeys in a round of 71 to finish on four under par.
Northern Ireland's four-time major winner Rory McIlroy had a better day with a par round following a three-over-par 73 in the second round.
American Jordan Spieth will take a one-shot lead into the final round of the Travelers Championship in Connecticut.