Shakers striker Ryan Lowe almost opened the scoring in the opening 15 minutes, but his chip was tipped over by Liam O'Brien.
Steve Kabba had an equally good chance for the Bees moments later, but Tom Lees was on hand to clear off the line.
That was as good as it got for the Bees, though, as Bury landed a double blow, Ajose rounding O'Brien after a mistake by Matty Parsons on 16 minutes, followed by a deft strike by Michael Jones five minutes later.
Although Barnet fought desperately to get back into the game in the second half, the Shakers' defence held firm and, indeed , they could have added to their tally on the hour mark when Ajose's shot was blocked on the line by Glen Southam.
But it did not matter as Bury saw out a win which leaves the Bees sitting in the League Two relegation zone.
Source: PA
Source: PA