For the Pilgrims victory was only their third in the last 15 game, while defeat for Cheltenham was their second reverse in nine outings since mid-February.
The match took fully 15 minutes to spark into any sort of life. But at that point Boston went ahead when Jason Lee got on the end of a left-sided Brad Maylett free-kick and buried a close range header past Shane Higgs.
The visitors almost levelled six minutes later when home keeper Nathan Abbey punched aside a fierce 25-yard free-kick by Brian Wilson.
Then back at the other end David Noble put Jermaine Easter clear, but Higgs dashed out to save at the Boston striker's feet.
Cheltenham threatened again in the 31st minute when Steven Gillespie threaded a pass wide to the overlapping Wilson, but his low angled drive was saved at full stretch by Abbey.
Boston went close to doubling their lead on the stroke of half time when a Maylett corner was met by Paul Ellender, whose header was held by Higgs.
Cheltenham exerted some early pressure in the second half and forced two corners, the second of which led to Shane Duff drilling an effort straight at Abbey.
Boston hit back and on 53 minutes Higgs had to save smartly from a Chris Holland volley following a cross from the right by Easter. And then in the 58th minute Courtney Pitt saw a 20-yard free-kick blocked by the visitors' wall.
Cheltenham tried to exert more pressure as the match entered its final throes and sent on substitutes Damien Spencer and Martin Devaney.
And, although both brought added creativity, they were unable to pierce a well organised home defence in which Ellender and Alan White were impressive.
Boston went 2-0 ahead on 88 minutes when Lee fired low past Higgs from 20 yards. But straightaway Cheltenham halved the deficit when Devaney netted from the spot after he had been tugged back in attempting to shoot.