The Pilgrims led through Marc Newsham's header midway through the second half and then saw Scott Garner denied from the penalty spot after Danny Hone had been sent off.
Substitute Ryan Kendall struck an 89th minute leveller to force a Tuesday night replay at The Jakemans Stadium.
It was almost the perfect awayday for the Pilgrims, but Adam Nicklin's superb penalty save to deny Garner and Kendall's last-gasp equaliser thwarted Dennis Greene's side.
Ferriby, who directed every attack through the giant figure of Tom Denton up front, had the game's early chances, with Newsham executing a crucial goalline clearance following Jamie Yates' testing free-kick.
Liam King's drive was subsequently deflected on to the post via Zak Mills' challenge, with Carl Piergianni hacking the loose ball to safety.
United went close when Mark Jones seized upon a mistake from Hone, only to be denied by a terrific recovering tackle from the Ferriby centre-half as a shot on goal beckoned.
Boston threatened at the beginning of the second period with Jonathan D'Laryea miraculously diverting Piergianni's close-range prod off the line following Greg Tempest's corner.
Yates' fierce free-kick bounced to safety off the Boston crossbar while King found himself unmarked in the Boston box, but dragged his shot across goal and wide.
United edged in front on 67 minutes when Jones dispossessed Yates and his cross-shot looped up for NEWSHAM to head home his second in two games.
The Pilgrims were presented with a glorious opportunity to kill the contest 10 minutes later when Hone brought down Jones in the box and was sent off, but Nicklin produced a brilliant save to his left to tip Garner's firm spot-kick on to the bar.
Ferriby changed things around, going 3-3-3 for the closing stages, and got their rewards when King was afforded too much space to cross from the right and KENDALL swivelled to fire home.
North Ferriby (3-5-2): Adam Nicklin; Mark Gray, Danny Hone, Josh Wilde; Jamie Yates (Gregg Anderson 80), Adam Bolder (Ryan Kendall 75), Liam King, Jonathan D'Laryea, Jason St Juste (Russell Fry 80); Tom Denton, Nathan Jarman. Subs (not used): Nathan Peat, Tom Nicholson (gk).
Boston (4-4-2): Joel Dixon; Liam Marrs, Zak Mills, Carl Piergianni, Rene Steer; Kaine Felix, Scott Garner, Kyle Dixon, Greg Tempest (Michael Hollingsworth 80); Marc Newsham (Jamie McGhee 83), Mark Jones. Subs (not used): David Fallah, Ollie Price, Sam Vince (gk).
Referee: Michael Salisbury.
Attendance: 321.
PIC: Eric Brown.