Jordan Thewlis set the ball rolling with a sixth minute opener and second half goals from Dominic Knowles, Thewlis again and Jay Rollins booked United's place in the third qualifying round draw.
The Pilgrims were good value for their win against Graham Drury's Daniels, whom United kept at bay for long spells at the Zeeco Stadium.
THEWLIS had already seen a fourth minute effort narrowly clear ex-Pilgrim Dan Haystead’s crossbar by the time he scored United’s opener two minutes later with a belting low finish following Knowles’ flick-on.
United continued to dominate and Haystead tipped a Walker free-kick behind for a corner while Rollins’ low drive was blocked wide of the near post by James Blunden.
Haystead made a fine save to thwart Walker before the break as the Daniels kept their cup hopes alive heading into half-time.
The hosts made three substitutions at the break and were lively for the opening minutes, with Tom Siddons glancing a header wide from Jon Challinor’s cross.
United were subsequently indebted to a fine Peter Crook save to scoop away Siddons’ close-range effort.
Haystead made another excellent stop to deny Walker before United added their second goal on the hour mark when KNOWLES worked himself a yard of space to bend a fine curling finish into the bottom corner of the net from 20 yards.
Rollins then rattled the inside of the post but the Pilgrims went three goals up with a quarter-of-an-hour to play when Walker robbed Sean Wright and THEWLIS ran on to steer the ball low to Haystead’s left.
Thewlis was denied his hat-trick by a Haystead save with four minutes to play but United were not to be thwarted for long, with ROLLINS walking home the rebound.
Stamford (4-3-3): Dan Haystead; Jordon Cooke (Jon Challinor 46), Sean Wright, James Blunden, Cleveland Taylor; Mike Armstrong (Pearson Mwanyongo 46), Rob Morgan, Harry Vince; Dec O’Hare (Ollie Brown-Hill 46), Tom Siddons, Cosmos Matwasa. Subs (not used): Ollie Luto, Jack Keeble, Ridley Braid, Joe Tarris (gk).
Boston (4-4-2): Peter Crook; Ben Middleton, Luke Shiels (Lewis Gibbens 83), Simon Ainge, Alex Whittle; Jay Rollins, Brad Abbott, Tom Platt, Nicky Walker (Jordan Adebayo-Smith 82); Dominic Knowles (Shaun Tuton 78), Jordan Thewlis. Subs (not used): Ashley Jackson, Andi Thanoj, George Green, George Willis (gk).
Referee: Callum Walchester.
Attendance: 671.