Newport's win over Accrington sees the Exiles leapfrog Pools, putting them two points in front with two games remaining.
Victory for Barnet, under caretaker boss Rossi Eames, was only their second win in 13 games.
But they controlled the match throughout against a side lacking in confidence, belief and ability.
John Akinde was denied an opener after 12 minutes when his angled header was tipped over for a corner.
From the set-piece Michael Nelson, the Barnet skipper, fired a shot against a post, while Jack Taylor's goal-bound shot was deflected wide.
Nathan Thomas got Pools' first shot on target after 64 minutes, failing to test goalkeeper Josh Vickers.
And the Bees went ahead in the 66th minute when Akinde turned and powerfully fired an angled drive past goalkeeper Joe Fryer.
Simeon Akinola made it two five minutes later, the Bees racing away down the left side and Jamal Campbell-Ryce setting up the substitute.
Akinde also missed a penalty, Fryer saving his weak spot kick with nine minutes remaining.
Match report supplied by the Press Association.
Match ends, Hartlepool United 0, Barnet 2.
Second Half ends, Hartlepool United 0, Barnet 2.
Attempt blocked. Padraig Amond (Hartlepool United) header from the left side of the six yard box is blocked.
Foul by Elliot Johnson (Barnet).
Carl Magnay (Hartlepool United) wins a free kick in the defensive half.
Attempt missed. Jamal Campbell-Ryce (Barnet) right footed shot from the left side of the box is close, but misses to the right.
Substitution, Barnet. Wesley Fonguck replaces Jack Taylor.
Charlie Clough (Barnet) wins a free kick in the attacking half.
Foul by Brad Walker (Hartlepool United).
Corner,  Hartlepool United. Conceded by Harry Taylor.
Attempt missed. Rhys Oates (Hartlepool United) header from the right side of the box is close, but misses to the right.
Attempt blocked. Jack Blackford (Hartlepool United) right footed shot from outside the box is blocked.
Attempt blocked. Nathan Thomas (Hartlepool United) right footed shot from outside the box is blocked.
Corner,  Barnet. Conceded by Matthew Bates.
Attempt blocked. John Akinde (Barnet) right footed shot from the right side of the box is blocked.
Penalty saved! John Akinde (Barnet) fails to capitalise on this great opportunity,  right footed shot saved  in the bottom left corner.
Penalty Barnet. John Akinde draws a foul in the penalty area.
Penalty conceded by Joe Fryer (Hartlepool United) after a foul in the penalty area.
Corner,  Hartlepool United. Conceded by Michael Nelson.
Foul by Tom Champion (Barnet).
Jack Blackford (Hartlepool United) wins a free kick on the left wing.
John Akinde (Barnet) wins a free kick in the defensive half.
Foul by Scott Harrison (Hartlepool United).
Attempt saved. Padraig Amond (Hartlepool United) header from the centre of the box is saved in the top centre of the goal.
Attempt missed. Nathan Thomas (Hartlepool United) right footed shot from the right side of the box misses to the right.
Attempt missed. Scott Harrison (Hartlepool United) header from very close range is close, but misses to the right following a corner.
Corner,  Hartlepool United. Conceded by Jack Taylor.
Goal!  Hartlepool United 0, Barnet 2. Simeon Akinola (Barnet) right footed shot from the centre of the box to the bottom left corner. Assisted by Jamal Campbell-Ryce.
Substitution, Hartlepool United. Jack Blackford replaces Nicky Featherstone.
Attempt missed. Jamal Campbell-Ryce (Barnet) right footed shot from the left side of the box is just a bit too high.
Goal!  Hartlepool United 0, Barnet 1. John Akinde (Barnet) right footed shot from the centre of the box to the top left corner. Assisted by Curtis Weston.
Substitution, Barnet. Simeon Akinola replaces Fumnaya Shomotun.
Substitution, Barnet. Tom Champion replaces Ruben Bover.
Attempt missed. Brad Walker (Hartlepool United) right footed shot from outside the box is just a bit too high.
Attempt saved. Nathan Thomas (Hartlepool United) right footed shot from outside the box is saved in the bottom left corner.
Substitution, Hartlepool United. Padraig Amond replaces Lewis Alessandra.
Foul by Jack Taylor (Barnet).
Rhys Oates (Hartlepool United) wins a free kick in the attacking half.
Elliot Johnson (Barnet) wins a free kick in the defensive half.
Foul by Carl Magnay (Hartlepool United).
Hartlepool dropped into the League Two relegation places as they slumped to a flat home defeat by Barnet.