Newsham opened the scoring in style, a minute after seeing a penalty saved by keeper Jose Veiga, only for City’s Daniel Nti to equalise four minutes before half-time after a defensive mix-up.
Weir-Daley grabbed the winning goal seven minutes after the break, although United were forced to survive a late barrage from City who saw Luke Hubbins denied on the line by a heroic Scott Garner clearance.
It was the first time United had recorded back-to-back home league wins since March 2012.
Showing an unchanged line-up for the fifth match in succession, the Pilgrims took the game to Worcester early on and Conor Marshall glanced a header wide following an Ian Ross corner.
City threatened for the first time when Danny Glover swivelled to fire the ball over the bar from 22 yards following George Williams’ long throw-in.
United were afforded a glorious opportunity to take the lead when Ricky Miller was fouled by Richard Munday in the box, but Veiga guessed correctly to deny Newsham from 12 yards.
There was no keeping NEWSHAM down though and a minute later he swept an angled drive beyond Veiga following Marshall’s delightful throughball.
Carl Piergianni headed over from a Ross corner, but City were soon level. Peter Bore surrendered possession outside the box, allowing Hubbins to cross for NTI to score one of the simplest goals he will ever net.
United upped the ante after the break and Netan Sansara’s driven cross was only inches away from being converted by Newsham at the back post.
The Pilgrims regained their lead on 52 minutes when Newsham played in Miller, with the ball looping up kindly for WEIR-DALEY to slam it home from close-range.
Veiga had to grab a Miller drive as the Pilgrims threatened to cut loose, but City soon had three great chances in a two-minute spell.
Nti rattled the post, before Ashley Walker clipped the top of the crossbar with his follow-up header, while Munday soon whistled a dipping volley a fraction too high from the edge of the box.
Newsham headed wide from a Weir-Daley cross, following good play from Miller, who was soon denied himself by Veiga.
United were indebted to a stoppage time goalline clearance from Garner, after Hubbins had wriggled free to fire goalwards.
Boston (4-3-3): Ashley Timms; Peter Bore, Scott Garner, Carl Piergianni, Netan Sansara; Ian Ross, Conor Marshall, Ben Milnes (James Reed 62); Spencer Weir-Daley, Marc Newsham (Ben Fairclough 79), Ricky Miller. Subs (not used): Stefan Galinski, Jordan Nuttell, Sam Vince (gk).
Worcester (4-4-2): Jose Veiga; George Williams, Graham Hutchison (Ashley Walker 25), Richard Munday (Danny Williams 80), Ellis Deeney; Luke Hubbins, Tom Thorley, Ebby Nelson-Addy (Ashley Sammons 59), Aaron Williams; Daniel Nti, Danny Glover. Subs (not used): Ethan Moore, Matt Sargeant (gk).
Referee: Lee Betts
Attendance: 919 (26 visitors)