York took the lead when Jake Hyde's shot cannoned off the right-hand post and Alessandra, on loan from Rochdale, pounced to net from 10 yards.
The visitors levelled just before half-time as John Akinde swept the ball home from James Pearson's right-wing cross.
Danny Galbraith and Luke Summerfield both tested Bees keeper Jamie Stephens but York were unable to find a winner.
The Minstermen are now seven points adrift of safety, following Hartlepool's victory over relegation-threatened rivals Dagenham & Redbridge.
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York City manager Jackie McNamara told BBC Radio York: "We deserved to win on chances created. I felt we could have got that second goal.
They got a lift and there's was very preventable. But, in saying that, we had a number of chances.
"The difference with fight and passion compared to last week was a lot better."
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Barnet manager Martin Allen told BBC Radio London: "It was always going to be a difficult game.
"I thought we looked a bit lacklustre first half, not quite up to the speed that we've been at, but in the second half I thought we played much, much better.
"I thought we had more opportunities where we perhaps could have won the game."
Lewis Alessandra's debut goal was not enough to stop League Two strugglers York being held to a draw by Barnet.