A stubborn 10th-wicket partnership from Adam Zampa (16) and Chris Green (34 not out) and a lightning-enforced delay combined to frustrate Tasmania in the final hour after they looked on top declaring 294 runs ahead.
But Webster produced two crucial late wickets to dash any hopes of NSW holding on for a draw.
The wickets came as Australia's selectors weighed up calling Webster into Australia's squad for the Adelaide Test, as cover for the sore Mitch Marsh.
The towering 30-year-old enticed Australian white-ball mainstay Zampa into nicking off to Jake Doran in the first ball of his new spell.
Paceman Jackson Bird edged behind to Tim Ward without scoring later that over, with the Blues (338, 239) all out only 13 minutes before play was due to close on day four.
Who else but Beau Webster!— cricket.com.au (@cricketcomau) Tasmania get their first W of the #SheffieldShield season! pic.twitter.com/EOqiQqAj9fNovember 27, 2024
The bowling heroics from reigning Sheffield Shield player of the series Webster (2-25) came after handy contributions of 61 and 49 with the bat and sealed a first win of the season for Tasmania (364, 8-268 declared).
"He's a special player at the moment," Tasmania captain Jordan Silk said of Webster.
"Any time I feel like we're in trouble I feel like I can throw him the ball and he just seems to be able to create something.
"I know there's a bit of talk around him potentially as a back-up (Test) allrounder and I'm very supportive of that move if they are to go that way."
The day began on a sombre note as tributes to Hughes brought the SCG to a standstill.
The former Test batter died 10 years ago on Wednesday after a bouncer struck his neck while batting in a Shield match at the famous ground.
The man who bowled that delivery, Sean Abbott, was consoled by NSW teammates as he wept during the pre-match minute's silence.
This morning at the SCG players and staff from NSW and Tasmania, along with Venues NSW staff and spectators, commemorated the life of Phillip Hughes with a minute's silence before play.— Cricket NSW (@CricketNSW) Always in our hearts, always 63 not out. pic.twitter.com/CbF67ZnjVoNovember 27, 2024
Abbott (4-71) went on to inspire a Tasmanian collapse of 4-10 early in the day, dispatching of Webster as the most crucial of his three wickets.
After the Tigers declared, Matthew Kuhnemann (3-76) made his mark by breaking a 61-run partnership between Oliver Davies (36) and Kurtis Patterson (33) that had the Blues on track.
Patterson then missed a chance to further reassert his international credentials when he skied Kieran Elliott, marking his first time in five innings he has not passed 50.
When Matthew Gilkes (11) was caught behind by Doran the next over off Kuhnemann, the Blues were wobbling at 5-120 with 175 more runs required for victory.
Josh Philippe (55) provided some late resistance before Mitch Owen bowled him on return to the attack, with Webster doing the rest in the final half hour.
