Jamie Stevens' goal seven minutes from time was enough to see off Chester and keep Boston's survival hopes alive.
His close-range header was enough to seal victory over 10-man City, who have now won only one of the last 12 games.
After a tame opening, Boston missed a golden chance to take the lead when Jamie Clarke saw an 18th-minute penalty pushed away by John Danby after Kevin Sandwith had handled on the line to keep out a Paul Ellender header - an offence that led to the City defender's dismissal.
Apart from that moment, the first half developed into a tame stalemate with only early off-target shots by Clarke and Francis Green for Boston threatening to break the deadlock - while at the other end little was seen of the Chester attack.
Boston went close in the 39th minute when Clarke drilled inches too high from 25 yards following a David Galbraith cross. And after 63 minutes Green got on the end of a Dave Farrell centre from the left but stabbed wide at full stretch.
A looping 10-yard header by Mark Greaves in the 69th minute from a Clarke corner came close to breaking the deadlock for Boston, but Chester continued to hold on.
However, Boston grabbed the lead after 83 minutes when Stevens rose at the near post to glance home a header from four yards from a Galbraith corner.
Boston almost doubled their lead two minutes later when, after a blocked effort by Green, the ball broke for Brad Maylett to shoot over from the edge of the box.