Lowry is five under par but still has four holes to play on Sunday before starting his final round at Oakmont.
American Andrew Landry is two shots behind with players catching up after storms affected the first day.
England's Lee Westwood is a shot back on two under, having played 15 holes, along with second-round leader Dustin Johnson and Spain's Sergio Garcia.
The leaderboard was constantly changing during an intriguing third round with Johnson, Lowry, Landry, Westwood and Garcia all having either the outright lead, or a share of it at some point.
None of the top seven on the leaderboard have ever won one of golf's four majors.
Lowry birdied the fourth, sixth and seventh to briefly move a shot clear at the top on his own.
But after taking a bogey-four on the eighth, he added birdies on the ninth and 12th to regain the lead and open up a slight advantage.
"I'm glad we didn't have to play another four holes as I was getting tired but I'm really looking forward to getting out there on Sunday," said world number 41 Lowry.
"I've never been in this position but this is where you want to be.
"My game is good and I feel very comfortable out there."
Johnson had begun the round with a one-stroke advantage and birdied the first hole to move two shots clear. He scrambled a par on the second after sending his tee shot, via a spectator's head, into a food stall.
However, the American double-bogeyed the third as his challenge faltered.
Westwood, 43, is looking to win his first major at his 73rd attempt. He held a share of the lead after chipping in for an eagle from 130 yards on the par-four fifth but finished the day on two-under par with three holes of his third round to play.
Garcia was also briefly a co-leader after hitting back-to-back birdies on the fifth and sixth as he bids for his first major after finishing second on four occasions.
Landry, who is ranked 624 in the world, led the American challenge with a birdie on the seventh giving him the outright lead and he ended the day as Lowry's closest challenger on three under with five to play.
South Africa's Branden Grace is the leader in the clubhouse after a four-under-par 66 in his third round.
World number one Jason Day, the 2015 PGA Championship winner, also hit a 66 after four birdies in his first five holes and picked up an eagle on the par-five fourth.
The 28-year-old Australian, who shot a six-over par 76 in his opening round, is six shots off the lead.
The third round will resume at 12:00 BST on Sunday, with the final round following later in the day.
Ireland's Shane Lowry holds a two-shot lead at the US Open with the leaders still to complete their third rounds.