Jennison Myrie-Williams should have put Vale ahead in the first half before Chris Neal saved twice from Andy Yiadom prior to 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 Barnet avoided a third straight defeat to climb out of the drop zone.
Leaders Vale, who have Gillingham hot on their heels, should have taken the lead after 35 minutes when Myrie-Williams shot wide on his unfavoured right foot from 12 yards.
Neal then did well to hang on to a deflected Yiadom strike before diving at the feet of the Barnet midfielder to keep the game scoreless at the break.
Oster forced Neal into another fine save at the start of the second half before Vale substitute Sam Morsy shot wide with 25 minutes to go.
Loft was then denied his first goal of the season by Stack in the Barnet goal and Pope hit a post.