Alex Reid's 20th minute strike was the difference between the sides at Grosvenor Vale as the Stones leapfrogged the Pilgrims in the league table.
United threw plenty at the hosts after the break - but Wealdstone held firm for a maximum three-point haul.
The Pilgrims started brightly and Jai Rowe forced keeper Dante Baptiste into a smart second minute save.
The hosts took the lead when REID seized upon some hesitant United defending to compose himself and roll the ball beyond the reach of goalkeeper Cameron Gregory.
Wealdstone kept Brad Nicholson's long throw opportunities to a minimum in the first half but from one half-cleared throw-in, Jordan Richards whistled a 20-yarder narrowly wide.
United went close through Jimmy Knowles nine minutes after the break, with Baptiste making a decent save, but the Accrington Stanley loanee was injured in the process and limped off nine minutes later to be replaced by James Gale.
The Pilgrims gave it their all in the closing stages, without unduly troubling Baptiste, and the closest either side came to a further goal saw Deon Woodman glancing a header narrowly wide from Max Kretzschmar's corner.
Wealdstone (3-5-2): Dante Baptiste; Enzio Boldewijn, Deon Woodman, Connor McAvoy; Will Randall, Kallum Cesay, Alex Dyer, Max Kretzschmar, Mustapha Carayol; Alex Reid, Tyler Walker. Subs (not used): Anthony Georgiou, Jed Brown, Christian Scott, Adrian Mariappa, Tobi Omole, Harrison Sohna, Luca Gunter (gk).
Boston (3-5-2): Cameron Gregory; Connor Teale, Zak Mills, Brad Nicholson; Jai Rowe, Dylan Hill (Adam Marriott 81), Jordan Richards (Greg Sloggett 67), Frankie Maguire (Keaton Ward 67), Cameron Green (Noah Wadsworth 81); Jimmy Knowles (James Gale 63), Jacob Hazel. Subs (not used): Jacob Scott, Jake Lovelace (gk).
Referee: Greg Rollason.
Attendance: 1,405 (233 visitors).
PIC: Chris Bray.