Watson's healthy lead from Friday was largely frittered away in the space of four holes on the front nine as he carded three bogeys, but the 2012 champion regained his composure to finish with a 74.
Having started the day at three-under for the championship, 20-year-old Spieth picked up two birdies and one bogey over the first nine holes, but found consecutive birdies at 14 and 15 to move into a share of the lead.
However, the chasing pack are in close touch with Spaniard Miguel Angel Jimenez carding the best round of the day (66) to move into contention to win his first major title.
Jimenez, 50, who would be the oldest champion at Augusta should he win, finished the third round two strokes behind Speith and Watson.
Defending champion Adam Scott, who started the day in contention, fell back after a round of 77.