The Canaries went ahead after just five minutes when Josh Sargent netted his 12th goal of the season, and his eighth in eight matches, and dominated for long periods.
But Oxford got an equaliser out of nowhere from Mark Harris 13 minutes later and went on to defend gallantly to secure a hard-fought draw that took them eight points clear of the drop zone.
Norwich moved to within three points of the top six - although they will regard this as a missed opportunity and two points dropped rather than one gained.
With Angus Gunn and George Long both out injured, young Chilean goalkeeper Vincente Reyes made his debut for the Canaries and was called into action after just 25 seconds to clutch a shot from Ole Romeny out of the air.
But Norwich looked the more dangerous side early on and were soon in front as Sargent maintained his hot streak in front of goal.
Oxford were caught out by a superb pass behind the left of defence from Callum Doyle and Kellen Fisher put in a precise low cross that was side-footed home from close range by the in-form American international.
The visitors quickly put the early setback behind them, equalising in the 18th minute as Will Vaulks' long throw caught Norwich out. The ball flicked off a defender into the path of Harris, whose shot on the turn from inside the six-yard box gave Reyes no chance.
Norwich dominated the remainder of the half without creating another clear chance, although Sargent did have a goal ruled out by a tight offside decision just before the break.
Ante Crnac shot straight at Jamie Cumming from a good position as the hosts maintained their forward momentum after the restart, with the Croatian frontman then seeing a fierce rising drive tipped over.
Norwich continued to pile on the pressure and with a minute of normal time remaining Borja Sainz burst clear on the left and, with a clear sight of goal, was denied by the outstretched leg of Cumming.
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