Max Gradel fired Leeds into a 13th-minute lead on Saturday as Simon Grayson's side enjoyed the better of the first half, but Canaries defender Barnett headed home David Fox's corner in the 65th minute to ensure a fourth straight Championship draw for his side.
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 last eight league matches, while Leeds have won three and drawn two of their games since successive home defeats last month left many questioning their play-off credentials.
City pair Fox and Andrew Surman - the latter in for Korey Smith and making his first appearance for Norwich since injuring his knee in August - both had early shots blocked, but Leeds snatched a 13th-minute lead with their first meaningful attack.
Last week's hat-trick hero Luciano Becchio was brought down on the edge of the penalty area by Barnett, but Hertfordshire referee Mick Russell played the advantage and the ball broke for Gradel, who dispatched his third goal of the season from 12 yards, low into the bottom corner.
Leeds enjoyed a period of sustained pressure after breaking the deadlock, but both sides struggled to carve out any clear openings until Gradel was denied a second goal by 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.
Norwich striker Grant Holt and midfielder Andrew Crofts both fired shots from the edge of the penalty area over the crossbar and Leeds defender Andy O'Brien blocked another effort from Holt as the home side burst out of the blocks at the start of the second half.
Leeds goalkeeper Kasper Schmeichel then denied Norwich an equaliser when saving superbly from Chris Martin, whose fierce 25-yard shot appeared destined for the top corner.
Leeds appeared to have ridden the storm and Norwich manager Paul Lambert sent on Wes Hoolahan and Simeon Jackson just after the hour mark in his search for an equaliser.
The change worked wonders as Norwich drew level three minutes later. Fox swung over a corner and defender Barnett rose above the Leeds defence to head home high into Schmeichel's goal from six yards.
Norwich swarmed forward in search of the winner and after Ward was brilliantly denied by Schmeichel's block in the closing stages, Crofts dragged his shot wide when well placed 12 yards out.