Roared on by huge crowds despite the inclement weather at Royal Portrush, Lowry had a nervy start but crucially holed a tricky putt to avoid dropping two shots at the opening hole.
The 23-year-old finished his tournament with an excellent three-under-par 68 to finish three-over for the championship, currently enough for tied-52nd.
The 32-year-old moved to five-under par for his round and 13-under for the tournament with birdie at the par-five 12th hole.
Koepka, who has not finished outside the top two in his last four majors, battled to a four-under-par 67 in the third round on Saturday to reach nine-under for the tournament, while Lowry delighted the home fans by powering to a 63.
His second straight 65 put him on 130 -- one stroke in front of fellow American Bill Haas who carded a 66.
England's Tommy Fleetwood and Lee Westwood are just one shot off the lead on seven-under.
Woods saw his bid for a 16th major title all but come to a premature end as he stuttered to a miserable seven-over 78, including six bogeys and a double bogey at the par-three sixth.