Jennison Myrie-Williams should have put Vale ahead in the first half before goalkeeper Chris Neal saved twice from Barnet's Andy Yiadom before the break. The Vale stopper then denied John Oster and, despite a late rally, neither Doug Loft nor Tom Pope could beat Graham Stack as the game finished goalless.
And Adams, whose side saw second-placed Gillingham close the gap with a win at Bradford, admitted his side were second best to Barnet.
"It was a good point based on our performance. Barnet were fantastic, they were better than us," said Adams. "We didn't have our usual spark, but we could've nicked it at times too. When you consider most of our best players were defenders, that says a lot about our performance.
"We gave Barnet too much time and respect on the ball. Edgar Davids ran the show at times along with (John) Oster. You can't give any type of player at this level that amount of time and space on the ball. We were very, very lucky to come in at half-time 0-0.
"We're not going to have it easy. Teams aren't going to roll over and let us tickle their bellies, because that's not how it is at this level. Barnet didn't set up to stop us, they just played their own way. We might have to look at the way teams play when we're away from home."
While Vale striker Pope hit the post with 20 minutes to go, Barnet were the better side for long periods and were unfortunate not to be ahead at the break. But with the Bees still locked in a battle to preserve their Football League lives, player-boss Davids admits they must be more clinical in attack.
"If you have a display of football like this against the top of the league than you can't ask for more," he said. "We played such great football and it's getting better and better. Hopefully the three points will come. Technically we were great and in attack we were great. We just weren't clinical enough.
"When you play against Port Vale it's hard because they're a team in form. You don't know what to expect. We could've had three points. We lack that clinical finish. We can play well but in the final third we have to score the goals. If you play well and it's 1-0 or 2-0 then you play more relaxed.
"Every game is a final for us and with Barnet you don't know what to expect. But I hope we can continue playing the good football we're playing. We're disappointed we've only picked up one point in our last three games, it's a big blow. "