Substitute Jeff Goulding scored the game's only goal 10 minutes from time at Whaddon Road but Cheltenham assistant manager Neil Howarth admitted it was former Barnet midfielder Sam Deering who helped secure the points.
The struggling visitors were awarded a penalty for handball midway through the second half but Robins keeper Scott Brown went the right way and palmed away Mark Byrne's spot kick.
Davids said: "We have to take the defeat on the chin and try and play better next time. We have played against a top side and you only have to slide a little bit and you can see why they are a good side.
"You can see positives but you also have to see your own mistakes and have to work on them because at the end of the day they are getting a bit annoying.
"There were little chances which we should have done better with. Everyone plays a lot of football in this league and the top side can lose against the bottom club - it is a very close league. That's the romantic side of this league and that's why I love it and don't underestimate it."