In an open 20 minutes period at the start of the match, both sides showed plenty of endeavour but didn't create any clear cut openings.
The hosts went close on 27 minutes as Luke Armstrong met a dangerous cross, but Scott Loach was equal to his effort as he made a fantastic reflex save.
Up the other end, it was the Bees who were next fashion an opening as JJ Hooper broke clear but couldn't find the finish as Ben Killip made a good save to deny him.
On 32 minutes it was the Pools who took the lead as a break down the left-hand side was eventually turned in from six yards by Armstrong.
It could have been two just a minute letter, as a carbon copy attack was this time turned over from almost the same position by Rhys Oates.
The hosts continued to press as the half drew to a close and thanks to Loach the score was kept at one, as he denied Gavan Holohan.
Tim Flowers' men looked threatening on the break but didn't create enough clear cut openings in the opening 45.
Into the second half and it was the hosts who created the first opening, as Holohan found space in the area for a shot, but a brave block from Liam Daly denied him. The Bees themselves were looking to use the pace of Petrasso on the breakaway, as they looked to forge a way back into the match.
Moments later it was Oates who went close as he flicked Jamie Sterry's cross just wide of the far post. Meanwhile, up the other end, Myles Judd's dangerous cross just evaded the sliding Hooper who was darting into the danger area.
The Pools continued to push throughout the half, but the Bees stood strong and managed to keep chances at a premium for the hosts who continued to find good areas out wide.
Second-half substitute Ep created the Bees clearest opening of the second half, as he came forward before finding Courtney Baker-Richardson, but the Barrow loanee's strike went over the crossbar.
As time ticked on Pools substitute Luke Molyneux fired a curling effort wide from just inside the area when he should have found the target after the dangerous Sterry played a great pull-back into the area.
The hosts had a golden chance to finish the game off with a minute left on the clock, but Armstrong somehow managed to miss despite having an open goal to tap home into.
The Bees tried to threaten as time ticked away but couldn't snatch an equaliser and ultimately left Victoria Park empty handed.
Barnet: Loach, Dunne (c), McQueen, Petrasso (Mason-Clark 76'), Hooper, H Taylor, Richards-Everton, Adeloye (Baker-Richardson 67'), Judd, R Taylor, Daly.
Subs not used: Azaze, McBurnie, Kefalas.