Goals from Glenn Rule and Joe Garvin did the damage, sandwiching Dayle Southwell's 59th minute equaliser, as County's new-look side kicked the season off in style.
United suffered a setback before kick-off with Zak Mills aggravating an injury in the warm-up and did not really get going until the early stages of the second half when Southwell eventually levelled.
Mills' absence saw Grant Roberts pressed into service at right-back with Peterborough United loanee Jack Friend being the late replacement from the bench.
Friend was soon in the action, heading wide at the back post following Kaine Felix's strong run and cross from the right. Southwell then undercooked a header from a deep left-wing cross and Danny Hurst comfortably gathered.
Stockport carved out a good chance when Everton loanee Callum Dyson waltzed through but cut inside and blazed wildly over.
Ex-Hatter Carl Piergianni was in the right place to nod Karl Ledsham's firm header off the line following Dyson's deep cross from the right but the hosts took the lead 10 minutes before the break.
United failed to deal with Jordan Thorniley's angled free-kick and Kayode Odejayi got the ball back across the box for RULE to head home.
There was almost an immediate response for United with Friend's back post drive rattling to safety off the woodwork.
The Pilgrims were a different beast after the break and should have levelled when Piergianni returned Kyle Dixon's partially-cleared corner and Mark Jones somehow fired wide at the back post with the goal at his mercy.
Piergianni's header flew a fraction wide following Dixon's flighted corner before United equalised when SOUTHWELL swooped to fire home after Stockport had failed to clear Dixon's deep cross.
League debutant Henrich Ravas kept United in front with a good save to deny Rule but he was beaten on 68 minutes when substitute GARVIN let fly from 30 yards with a low drive into the bottom corner.
United huffed and puffed in pursuit off second leveller and the closest they came was in the 85th minute when Hurst excelled to keep out Jay Rollins' blockbuster.
Stockport (4-3-3): Danny Hurst; Danny Morton, Sean O'Hanlon, Jordan Thorniley, Gareth Roberts; Glenn Rule, Lewis Montrose, Karl Ledsham (Ryan Ellison 88); Callum Dyson (Joe Garvin 63), Kayode Odejayi (Lamin Colley 79), Sefton Gonzales. Subs (not used): Andy Robinson, Paul Tyson (gk).
Boston (4-4-2): Henrich Ravas; Grant Roberts, Scott Garner, Carl Piergianni, Liam Marrs; Kaine Felix, Kyle Dixon, Jordan Deacey (Jamie McGhee 55), Jack Friend (Jay Rollins 49); Dayle Southwell, Mark Jones. Subs (not used): Jake Sheridan, Cameron Johnson, Jason Alexander (gk).
Referee: Andrew Miller.
Attendance: 3,396 (179 visitors).