Max Gradel fired Leeds into a 13th-minute lead as Simon Grayson's side enjoyed the better of the first half, but Barnett headed in David Fox's corner in the 65th minute to reward a spirited second-half showing.
Norwich dominated the second period and had the visitors holding on for a point at the end, but have now won only one of their past eight league matches, while Leeds are now unbeaten in five.
After the match, the Norwich manager, Paul Lambert, whose side have now drawn four in a row, said: "It's another draw but it's not frustrating because I know how well the team are playing. We created chance after chance and on another day could easily have won."
Fox and Andrew Surman both had early shots blocked for the home side, before Leeds took the lead with their first meaningful attack.
Luciano Becchio was brought down on the edge of the penalty area by Barnett, but the referee Mick Russell played the advantage and the ball broke for Gradel, who struck his third goal of the season from 12 yards, low into the bottom corner.
Leeds enjoyed a period of sustained pressure after scoring, but both sides struggled to carve out any clear openings until Gradel was denied a second goal by the Norwich goalkeeper John Ruddy, who tipped over the Ivory Coast international's 25-yard shot.
Gradel then fired in a fierce drive that was well held by Ruddy and saw another effort blocked by Norwich defender Elliott Ward shortly before half-time.
The Norwich striker Grant Holt and the midfielder Andrew Crofts both fired shots from the edge of the penalty area over the crossbar and the Leeds defender Andy O'Brien blocked another effort from Holt as the home side began the second half brightly.
Leeds appeared to have ridden the storm and Lambert sent on Wes Hoolahan and Simeon Jackson just after the hour to strengthen his attack. The change worked: three minutes later Norwich were level. Fox swung over a corner and Barnett headed home from six yards.
Grayson felt the Canaries' equaliser should not have stood. "We are extremely disappointed with their goal because it was a blatant push by Chris Martin on Kasper Schmeichel," he said.
Norwich were the stronger side from then but could not find the winner.