The Dubliner, 36, found three late birdies to card a three-under par 69 on a fascinating third day and move into a nine-man tie for seventh place on two-under par.
South African Immelman birdied the 18th with a glorious approach to two feet for a 69 that doubled his overnight advantage to two shots from Nashville's Brad Snedeker on 11-under par.
Snedeker also birdied the last for a 70 to join Immelman in the final group with left hander Steve Flesch a shot further back on eight under after a 69.
England's Paul Casey is four behind in seven-under par with Tiger Woods facing a six-shot deficit after a bogey-free 68 moved him to five-under.
But with bad weather forecast for Sunday, Harrington is not giving up hope of contending on the back nine.
Harrington said: “I suppose I'm reasonably happy with the score and feel like I maybe should have done a little bit better. Early on in the round things weren't happening, so it was nice to make three birdies in the last four holes.
"I am pleased, but certainly the opportunity was there today to shoot a good score. Looking at the guys on the leaderboard at this very moment, I'm wondering how far away from me they're going to be at the end of the day.
"Who knows, I hear tomorrow is going to be a blustery day, colder day, so it certainly could make it awkward for people out there leading."
The Dubliner started the day tied for 21st on one-over par before cruising through the turn in red figures after balancing a bogey at the fourth with birdies at the par-five second and eighth.
A bogey at the par-five 13th, where he dumped his four-iron approach from a sidehill lie in Rae’s Creek when he had just 175 yards to the front, looked to have put paid to his chances.
But he finished well and while he was disappointed not to have gone even lower, he knows that victory is not out of the question.
Reflecting on the 13th, he said: "If I look back at anything in the round, that was a couple of shots gone. That would have made it a very sweet round actually if I had birdied 13.
"I took plenty of club, but off the downslope, grain into me, I tried to help it a bit up in the air and caught it fat.
“I had only 175 yards to the front of the green. They do know how to make every shot difficult out there.”
"But you know, this golf course, you can't automatically say I'm going to go all-out with any pin position. You have to wait and see where you are, what club you have and whether you feel good about the shot and choose the right shots at the right time.
“I'd be happy if I can get myself into that position with nine holes to go tomorrow and have that feeling coming down the back nine.
“I always look back at how I prepared and how I got through the week. But as regards results, I have no interest in finishing anywhere else in the field but first.”
Woods, who has triumphed nine times in 11 starts worldwide since last August, accepts he faces an uphill task in Sunday's final round with cold weather and strong winds forecast.
"If everything holds up and we get the weather we're supposed to get tomorrow, you just got to hang in there and hang around," he said. "You know that anything can happen.
"There's not many guys ahead of me but it all depends on what they do, how low they go. Hopefully, I can stay within striking distance tomorrow."
For Woods to clinch his fifth green jacket, he will have to enter uncharted territory, having never before come from behind after 54 holes to win a major championship.
Harrington knows that anything is still possible and has no intention of throwing in the towel.
He said: “This is Augusta. If it was anywhere else, we wouldn't even be considering it, but this is Augusta.”