Sandhu, who claimed figures of 5-35 in just his second match, removed dangerman Tim Paine (30 from 44) and ripped through the Tigers tail after in-form opener David Dawson had struck a maiden one-day century in the Blues' 4-246.
Spin twins Luke Doran (3-64) and Adam Zampa (1-44) had orchestrated a Tigers' collapse with the home side needing a bonus point win to remain a mathematical chance of playing in the February 27 decider.
Doran bowled Tasmanian opener Jonathan Wells for a duck in the second over before removing Alex Doolan (21 from 19) and Jason Krejza (7 from 9).
It was leg-spinner Zampa, though, that snared the big one when he had Ricky Ponting caught behind after the former Australian skipper had looked ominous, cracking five fours in his 28 from 29 balls.
For Tasmania, only Jordan Silk (54 from 71) and Evan Gulbis (45 from 54) offered much resistance.
Tasmania stay fifth with the loss and NSW sixth despite their first one-day victory since September 16 last year.
The Tigers made a bright start, reaching 50 off just 61 balls despite the loss of Wells at 1-6.
Ponting departed at 2-58 and Paine without a run being added, caught behind off the impressive bowling of 19-year-old Sandhu.
Any chance of a bonus point looked gone when Doolan went at 4-101, and it was 5-119 when Daniel Hughes took an unusual catch that had ricocheted off the bowler's hands to dismiss Matt Johnston (8 from 20).
Sandhu took the last four wickets, including Silk's, to mop up the tail, the Tigers all out for 209.
Earlier, Dawson continued to torment his former state with an unbeaten 124 from 139 balls.
The 30-year-old put on 93 with Tim Cruickshank (61 from 85) for the second wicket to set up NSW's total after they had won the toss.
He made 75 in the first innings of last week's drawn Sheffield Shield match but his unbeaten 19 in the second was just as important as he saw off the final over of the tense draw.