But Barnet's goal scorer Adam Birchall was left to rue a great chance five minutes from time when he failed to beat stranded keeper Scott Loach with a weak lob.
City were booed off at the break but they were roared on by another large crowd after Johnson headed their equaliser from close range with 15 minutes to go.
In a frantic finale substitute strikers Luke Medley and Willy Topp got in each others' way when either could have scored the winner for City, but Barnet will be wondering how they squandered the three points.
City made a lively start on the back of two successive home wins, but the fizz was knocked out of them by some woeful passing and hesitant defending.
Loach made amends for an early mix-up with centre-half Mark Bower with a vital block after 17 minutes when the impressive Birchall shrugged off Ben Starosta's weak challenge on the edge of the box.
Six minutes later Loach was left flat-footed when a fierce 30-yard shot from Anthony Thomas rebounded back into play off the City bar and Birchall did well to control the loose ball and roll it into the Bradford net from 16 yards out.
City rarely threatened in the opening period and Loach had to rescue them again just before the break when he stood up well to Thomas as he burst into the box after another defensive lapse.
City boss Stuart McCall pepped up his attack by bringing on the speedy Medley, but it was Johnson, who had dropped back into midfield, who grabbed the leveller.
Winger Joe Colbeck ran himself ragged in the City cause and it was his pin-point corner which gave Johnson the chance to nod home his fourth goal of the season.
With ten minutes to go Barry Conlon's cross should have been tucked away by either Medley or Topp, but the City duo conspired to put each other off with the goal at their mercy.
In an end-to-end finish Birchall burst clear after beating City's offside trap, but he failed to get enough on his lob from the edge of the box and Loach gratefully grabbed the ball.