The Bees came from behind three times against promotion contenders Bradford City and fully deserved at least a draw, but it took a superb save from Bantams keeper Rhys Evans to deny Michael Leary a stunning winner with just eight minutes left.
Bradford were their own worst enemy as they gifted Barnet a way back into the contest after in-form Barry Conlon had headed City in front with his fifth goal of the season after just ten minutes.
City's nervous defence had already survived a let-off when Evans saved brilliantly after Barnet striker John O'Flynn raced clean through.
But the home side failed to tighten up and, after 20 minutes, a poor back-pass from Thomas Moncur put Evans under pressure and the keeper's miscontrol gave O'Flynn an easy tap-in into an empty net.
Stuart McCall's men, who could have gone top of the table with a win, hit back on the half-hour when striker Peter Thorne guided home Paul McLaren's free-kick from close range, but again they failed to hold the lead for long.
A great run by O'Flynn set up a chance for Adomah after 31 minutes, and when Evans did well to save his shot, team-mate Nicky Nicolau was on hand to drill home the equaliser with a low 16-yard shot.
Conlon, who has hit top form in the last few games, restored City's advantage within a minute after he controlled a throughball before firing a right-foot shot beyond the stranded Lee Harrison.
Bradford were relieved to go in 3-2 ahead, but it was Barnet who started to stepped up their game after the break.
Adomah and Nicolau had the Bantams back-peddling and it seemed only a matter of time before they grabbed the equaliser again.
Harrison was happy to see the ball bounce off his bar after he pushed away a Nicky Law shot and the escape proved decisive 14 minutes from time.
Adomah was quickest to react to Adam Birchall's touch in the Bradford box and, although Evans protested long and hard that the winger's eight-yard shot had not crossed the line, referee Nigel Miller gave it the nod after consulting his linesman.
Adomah headed wide soon after and Evans pulled off his stunning late save to ensure City salvaged at least a point from a game they looked like throwing away.