Halfway through their pink-ball clash, the home side was 1-81 in its second innings and boasts an overall lead of 226.
Third-placed NSW must beat Tasmania, who are last, in the final round to leapfrog Queensland and face SA in the final.
While Queensland are in major trouble after SA took control of their game in Adelaide, NSW will have to rally in Hobart if they are to take advantage of their opportunity.
NSW struggled in the middle session of the day-night game on day two and were rolled for 186 in their first innings, as Webster and Meredith excelled with the ball.
Weatherald then ended the day with an unbeaten 63 and his dismissal early on Monday will be a priority for the NSW attack.
Tasmania resumed on 7-309 and the early interest on Sunday afternoon was whether captain Jordan Silk, in his 100th Shield game, could score a century.
He delighted the home crowd by reaching 112 in 208 balls, with 12 fours, before Ben Dwarshuis bowled him to end their first innings.
Sean Abbott led the NSW attack with 6-99 and Dwarshuis snared 3-77.
Bowling with significant pace, Meredith took two wickets in an over as the visitors were put on the back foot in their first innings.
Test opener Sam Konstas only made five and while Kurtis Patterson (33), Josh Philippe (25) and Abbott (30) made starts, NSW lost 3-0 to stagger to 7-134.
But Dwarshuis ensured Tasmania would bat again on Sunday, belting 40 from 31 balls, with six fours and two sixes, to help take NSW past the follow-on target.
Webster, the Test allrounder, took 5-44 and Meredit claimed 3-45.
Jackson Bird dismissed Nivethan Radhakrishnan for six to give NSW the early breakthrough they needed in the Tasmanian second innings.
Weatherald and Tim Ward (11no) then batted through to stumps, turning the screws tighter on NSW.
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