The Pilgrims had trailed to Amari Morgan-Smith's 37th minute penalty and were subsequently indebted to a pair of brilliant Peter Crook saves either side of half-time to keep themselves in contention.
Wright struck United's leveller in the 62nd minute but the Pilgrims were unable to find a winning goal, despite dominance of the game thereafter.
The Pilgrims, who were completely unchanged for the first away fixture of the season, made the early running and saw a Wright effort loop wide off Dominic Smith.
Jordan Thewlis was soon caught in two minds between shooting and crossing following terrific play from Tom Platt, Alex Whittle and Andi Thanoj and the chance went begging.
Home keeper Jack Atkinson made a brilliant save to keep out a stinging Jay Rollins drive while Platt’s glancing header landed on the roof of the net following Whittle’s cross from the right.
The Reds edged in front eight minutes before half-time when MORGAN-SMITH was bundled over by Ben Middleton before picking himself up and firmly dispatching the spot-kick awarded by referee Scott Tallis.
Platt smashed a ferocious drive a fraction too high in response but the Pilgrims were grateful to Crook for making a super stop to keep out Morgan-Smith just before half-time.
Brad Abbott and Thewlis were both narrowly off target with efforts from distance in first half stoppage time.
Crook pulled off another fine save four minutes after the break when he somehow kept out Smith’s firm drive from close-range.
The United keeper turned provider when his opposite number Atkinson misjudged a long kick forward and WRIGHT was able to run on and steer home an angled effort which nestled into the net via both posts.
United cranked up the pressure and could have taken the lead when former Alfreton favourite Luke Shiels hooked an effort off target following Whittle’s corner.
Despite dominating territorially, United were unable to muster too many efforts to trouble Atkinson in the Alfreton goal.
Alfreton (3-5-2): Jack Atkinson; Dominic Smith, Ryan Qualter, Josh Thacker; Josh Clackstone (Nyle Blake 89), Dale Whitham (Bobby Johnson 46), David Lynch, Danny East, Josh Wilde; Danny Clarke, Amari Morgan-Smith. Subs (not used): Ben Tomlinson, Shane Killock, Declan Bacon.
Boston (4-3-3): Peter Crook; Ben Middleton, Luke Shiels, Simon Ainge, Alex Whittle; Brad Abbott, Andi Thanoj (Nicky Walker 79), Tom Platt; Jay Rollins, Jake Wright (Shaun Tuton 72), Jordan Thewlis. Subs (not used): Ashley Jackson, George Green, George Willis (gk).
Referee: Scott Tallis.