'I haven't won enough to think that it certainly comes easy, so I know how important it is,' said Austin, 43, who has three career PGA TOUR victories.
The birdie gave the team a final-round 9-under 63 in the scramble format and a 29-under 187 total.
'I had a good vibe for him on that last putt,' said Calcavecchia, who putted first and gave his partner a good read.
Norman and Watson both just missed birdie tries from 25 feet after Austin made his putt. Norman's ball ended up inches right of the hole, and Watson's putt ran out of gas and broke in front of the hole to the left.
'Am I disappointed for myself? Yeah. But more for Bubba,' Norman said. 'I would love for him to get a victory. To get a taste of it.'
Norman was trying to win for the first time since the 2001 Skins Game, while Watson hasn't won as a professional. They tied the Shootout with a two-putt birdie on the par-5 17th, while Austin and Calcavecchia parred the hole after poor tee shots.
'Of course I'm disappointed,' Norman said. 'We all play to win the championship, no matter what the championship is. We put ourselves in position.'
When Scott McCarron and Steve Elkington birdied the 18th for a 62, it created a five-way tie for third with defending champions Jerry Kelly and Rod Pampling, Scott Hoch and Kenny Perry, Chris DiMarco and Camilo Villegas and first-round co-leaders Fred Funk and Jeff Sluman.
Norman, who has won his own event just once, and Watson trailed most of the final round, but tied it with the birdie on 17. Austin and Calcavecchia both put their drives in a fairway bunker on the par-5, and both ended up going over the green. They made a par, the only team in the field to do so in Sunday's scramble format.
Austin and Calcavecchia missed short birdie putts on Nos. 15 and 16, and a makable birdie putt on 17.
'We were struggling, and then they started struggling at the end and started missing some putts, and kept giving us chances,' Watson said.
Austin was playing in his first-ever challenge season event. Calcavecchia won the tournament for the second time; he and Elkington did it in 1995.
Austin and Calcavecchia split $700,000 of the $2.8 million purse. The 12 two-man teams played modified alternate shot on Friday and best ball on Saturday.