Irwin closed with an even-par round of 71 to end the event at 8-under-par 276. Haas, making his Champions Tour debut, posted a 1-under 70 to end at minus-7.
'Winning this tournament is one of those dream things,' said Irwin, who collected $360,000 for the win. 'You lay awake at night, like I did last night, kind of thinking about it.'
The Monday finish was caused by rain that delayed the tournament each of the five days action was contested. That forced the first Monday finish of this event since the 2000 playing at PGA National in Palm Beach Gardens, Florida. When play resumed Monday, players battled winds that blew in excess of 25 miles per hour.
Craig Stadler, who hit a poor second shot to the par-5 18th, pitched in for eagle to close a round of 2-under 69 and end the event in third place at 5-under-par 279.
Canadian Dave Barr, Englishman Mark James and 2001 Senior PGA Championship winner Tom Watson shared fourth place at 2-under-par 282. Mark McNulty and Gil Morgan ended one stroke further back at minus-1.
The final round duel between Irwin, the Champions Tour's all-time wins leader now with 40, and Haas, playing in his first Champions event, was tight throughout. Haas never led, while Irwin's biggest cushion was two strokes and they were twice tied.
Irwin bogeyed the first hole on Sunday afternoon before play was suspended for the night. When they returned to the course on Monday, Haas faltered to his only bogey of the round at the third to fall to minus-5, two behind Irwin.
The 58-year-old Irwin parred his first three holes on Monday before draining a birdie at the fifth. That gave him a two-stroke lead over Barr and Watson. Irwin ran off four straight pars to head to the back nine up by two shots.
Irwin, who won this event three straight years from 1996 to 1998, matched Haas' birdie at the par-5 10th. Things went downhill though for Irwin.
He bogeyed the 11th to fall to minus-8. Irwin's tee shot off the 12th tee found the right rough and he was forced to lay-up. He pitched his third onto the green, but was unable to make the par saving putt. That dropped him to minus-7, where Haas joined him in the lead.
At the very next hole, Irwin got a favorable ruling. His tee shot ended on the edge of the first cut of rough, but an official ruled he could take advantage of the lift, clean and place rules that were in affect saying his ball was touching the fairway.
Irwin cleaned his ball then dropped his second shot within eight feet and he drained that birdie to retake the lead. Over the next three holes, Irwin two- putted for par on each to maintain his lead.
The 1998 and 2000 U.S. Senior Open champion dropped his second shot to the par-4 17th in a greenside bunker. Irwin blasted his third onto the green, but could not sink the 12-foot par putt.
Heading to the final hole with Haas playing in the group ahead of him, Irwin found the fairway off the tee. He watched from the fairway as Haas, who has never won a major, two-putted for par.
Irwin, needing birdie to win, knocked his second shot on the green. Despite having a difficult putt over a slope, Irwin's eagle try came up just short and right of the cup. He calmly tapped in his birdie putt for the win.
'I can't say I was happy with what I was looking at,' said Irwin of his eagle putt. 'Fortunately, I read it well. I'm proud, I'm relieved and I'm glad it's over. It's been an awkward week for everyone.'
Irwin now has won at least two Champions Tour events in each of the last 10 years. He also claimed his seventh Champions Tour major.
Haas, who will return to the PGA Tour for The Memorial later this week, parred the final eight holes. In doing so, he became the only player to complete the final nine holes of the event without a bogey.
'That's disappointing that I didn't get that done,' said Haas, who owns nine PGA Tour wins. 'I just hit a very poor putt, probably one of the worst shots I hit all day. I had as good an opportunity as I'm going to have.'
John Harris struggled to a 5-over 76 to finish the tournament at even-par 284. It was his first career top-10 finish on the Champions Tour in 32 events.
Wayne Levi, Jerry Pate and Raymond Floyd shared 10th place at 1-over-par 285. Allen Doyle, the 1999 Senior PGA Champion, was one stroke further back at plus-2. He was joined there by Andy Bean, Tom Jenkins and R.W. Eaks.