John McGrath headed the Brewers in front just before half-time but Jake Speight rescued a point for the Bantams after 72 minutes, slotting home at the back post after Albion had failed to clear a long throw. City are now seven points clear of the relegation zone.
Jackson said: "It was a big relief. It was such high pressure to go one down and I thought the players showed great character to come back and equalise. We could have won it at the end and we could have lost it as well. I am just delighted to get a point.
"It was critical we didn't get beaten. I have got to be pleased with the result. It was a big night for the club. It would have been nice to get the win but I think Barnet, who are next to the bottom, need to win three of their next four matches now.
"I am confident going to Accrington on Saturday to get something out of the game, but I believe we will win one more game before the end of the season."
On Speight, Jackson added: "I think Jake Speight livened things up when he came on."
Source: PA
Source: PA