United fluffed a hatful of chances with new signing Rob Collins the chief culprit.
The ex-Crawley striker raced clear of the Grantham defence on 21 minutes but scuffed his finish wide. The miss was made to look even worse just a couple of minutes later when the home side took the lead through Stuart Wilson.
Collins had another great opportunity early in the second half, hitting the post from six yards. United struck the woodwork for a second time when Landry Zahana-Oni's header came back off the crossbar and bounced down in front of the goalline.
Grantham doubled their advantage on 68 minutes when Jon Hawley converted a penalty awarded for a foul on Gary Bull by keeper Dave King.
United pulled a goal back late on when Joel Bevis fired home but they were unable to make any more headway.