Thanoj's 38th minute free-kick, in his 76th appearance for the club, sealed United's progress to the competition proper for the first time as a non-league club since 1997.
The Pilgrims, who were hampered by injuries and illnesses ahead of kick-off, subsequently lost Mansfield Town loanee Lewis Gibbens to a groin injury after only seven minutes.
Gibbens started in the place of injured captain Luke Shiels, while top goalscorer Jordan Thewlis was deemed only fit enough for the bench.
Gibbens injured his groin in a fourth minute clash with Danny Glover and hobbled off three minutes later to be replaced Ben Middleton in another re-jig to the United backline.
Brad Abbott tested Town keeper Daniel Platt before the Pitmen had three golden chances to take the lead.
Glover’s free-kick was blocked, with James Hurst blasting the rebound wide, while Peter Crook made fine saves to deny Tom Elliott’s free-kick and Osebi Abadaki from distance.
United’s breakthrough arrived seven minutes before half-time when THANOJ stepped up to curl a 20-yard free-kick beyond the reach of Platt following Ben Bailey’s foul on Abbott.
Hednesford wanted a penalty within the first minute of the second half, but referee Dale Wootton was unmoved following Martyn Woolford’s possible handball.
United sought a killer second goal and Dominic Knowles freed Jay Rollins down the United right, but the Pilgrims’ longest-serving player crashed an angled effort high and wide.
Boston managed the game well during the second half and Hednesford’s chances were limited, with Elliott firing off target from distance and substitute Jonathan Wafula providing two telling last-gasp interventions to see the Pilgrims safely through.
Hednesford (4-2-3-1): Daniel Platt; Matt Curley, Kieran Morris, Ben Bailey, Sam Griffiths (Tom Fry 45); Reece King, James Hurst; Isak Reid, Tom Elliott, Osebi Abadaki; Danny Glover (Kieran Wells 77). Subs (not used): Charlie Gatter, Jack Hallahan, Kristian Scott, Billy Daniels, Jose Veiga (gk).
Boston (4-3-3): Peter Crook; Ashley Jackson, Simon Ainge, Lewis Gibbens (Ben Middleton 7), Alex Whittle; Brad Abbott, Andi Thanoj, Tom Platt; Jay Rollins, Dominic Knowles (Jordan Thewlis 90), Martyn Woolford (Jonathan Wafula 80). Subs (not used): Nicky Walker, Jordan Adebayo-Smith, Shaun Tuton, George Willis (gk).
Referee: Dale Wootton.
Attendance: 891 (283 visitors).