Two goals from Jake Wright inside the opening 25 minutes of the quarter-final clash put the higher level Pilgrims in control.
A battling Daniels display was then dented in the final minute when Simon Ainge headed home the Pilgrims' third goal.
Boston, as expected for a side who play two levels higher than the Daniels, controlled the opening exchanges.
Jordan Thewlis flashed an early effort wide while Stamford goalkeeper Dan Haystead had to react swiftly to turn away a fierce Jake Wright shot at his near post.
Stamford had competed well in the first 15 minutes after an energetic opening, but fell behind shortly after.
Jay Rollins produced a smart run down the right and his cross picked out Wright whose first-time finish flew past Haystead.
The Pilgrims then doubled their lead seven minutes later as Alex Whittle's surging foray into the box saw him pick out Wright who converted his and Boston's second goal.
Wright came close to completing his hat-trick from another Whittle cross, but this time Josh Egginton got in a vital block to divert the goalbound shot over the crossbar.
Stamford's best chance came in the latter stages of the first half as James Hicks took advantage of a misunderstanding in the home defence, but his low shot was superbly blocked by Boston goalkeeper Peter Crook.
Boston started the second half on the offensive again with Haystead making a fine save to deny the marauding Whittle while Matt Dawson produced a vital headed clearance to deny Rollins converting a Thewlis cross.
Stamford substitute Tom Siddons was then denied by a brilliant full length save from replacement Boston goalkeeper George Willis after a sublime header from a free-kick.
Boston were forced to play the final 10 minutes a man short after George Green was taken off injured with the Pilgrims having already used all of their substitutes, but Stamford were unable to find a way through and Ainge headed home the hosts' third late on following a corner.