Adams believes the Valiants will become a favoured scalp in the division after they moved above Martin Allen's Gills with a 2-1 win at Priestfield.
Lee Hughes claimed the winner on his debut for Vale, although the final touch may have come off home defender Adam Barrett.
Hughes' goal made it 2-0 in the 20th minute following a rip-roaring start from Vale, with Tom Pope having headed in his 24th league goal of the season early on.
Danny Kedwell pulled one back less than 60 seconds after Hughes' finish, but the Gills were unable to force an equaliser.
Adams is now urging his side to avoid any complacency. He said: "We've shown once again that away from home we're more than capable of getting results, but we have to keep reminding ourselves of how we got them.
"All the principles and disciplines you need to get results we've applied them. I think it was a battling performance to a man; they put their bodies on the line.
"Racing into a two-goal lead, it looked as though we were comfortable at that stage but Gillingham don't give you a minute's peace, you can see why they've been top of the table for so long.
"It feels great being top. It's doubly hard now - we're there to be shot at.
"We've experienced it before this season, and in my first spell at the club we didn't handle it too well. I told the boys to not get carried away - it's only three points, but they showed they can compete with the best."