Four times they saw efforts hit the Cheltenham goal frame in addition to seeing another shot cleared off the line and keeper Shane Higgs excel on several occasions.
Higgs had to be alert twice in the opening stages. First he pushed away an 11th minute close range effort by Julian Joachim and 60 seconds later went full stretch to tip over Alan White's header from a Danny Thomas free kick on the left.
Boston went close again in the 20th minute when more pressure down the left led to Thomas crossing for Joachim and his shot deflected wide off the legs of Higgs.
Another chance arrived for Boston on 30 minutes following more good work by Thomas. This time Stewart Talbot saw a chip shot diverted over the bar off defender Shane Duff.
Apart from Kayode Odejayi heading wide from a 26 minute cross by John Melligan, little was seen of the visitors attack.
The game sprang to life four minutes before the break - first when Odejai burst clear and forced Michels Kuipers into an excellent save and then at the other end when Joachim's surging run and shot was thwarted by Duff's goal-line clearance.
Cheltenham, though, continued to live a little uneasily in defence. Three minutes into the second half Francis Green burst clear only to be denied by more Higgs saving strengths. Then in the 51st minute Green cut in from the left only to see his low angled shot bounce away off the far post.
Cheltenham went close when a 68th minute 20-yard free kick by Brian Wilson was punched clear by Kuipers. But it was at the other end where most of the pressure was mounting - Boston again being out of luck when a Chris Holland cross shot rattled away off the inside of a post.
Sensationally the same upright rescued Cheltenham again five minutes later from a Lawrie Dudfield shot, with Mark Greaves remarkably heading the rebound against the visitors bar.
Late in the piece Cheltenham almost snatched an 88th minute winner when substitute Damien Spencer wriggled into a shooting position but saw his effort skim wide.