The Pilgrims overcame a stubborn and youthful Grimsby Town side at The Jakemans Stadium and were indebted to keeper George Willis for saving Michael Davies' penalty at 0-0.
Chapman opened the scoring from the spot for United and Beatson and Tshimanga added second half strikes to seal a meeting with Gainsborough Trinity later in the season.
Boston (4-2-3-1): George Willis; Taron Hare (Jack Broadhead 57), Jack Cowgill, Jordan Keane, Brad Beatson (Jan Yeomans 72); Jamie McGuire (Tom Batchelor 72), Adam Chapman; Ashley Hemmings (Mason Warren 77), Harry Vince (James Clifton 57), Tyrell Waite (Kabongo Tshimanga 57); Karl Hawley (Gregg Smith 57).
Referee: Alistair Wilson.
Attendance: 677 (152 visitors).