Either side could have won a very defensive game at the County Ground but there was little of note in the first half except for a long-range Wes Hoolahan shot in the first minute.
Swindon's best effort was a low driven cross by Charlie Austin which was hacked away and there was a strong appeal for a free-kick on the edge of the penalty area when Frank Nouble seemed to be impeded by Michael Nelson.
It all changed soon after the break when Norwich broke the deadlock in the 51st minute. Hoolahan did well in midfield to release Mike Rose on the left and from his cross Grant Holt headed powerfully home from close range.
Norwich were having the better of the exchanges and Chris Martin went close to increasing the lead when his 30-yard free-kick clipped the crossbar.
They looked set to take all three points but in injury time Jonathan Douglas' strike was brilliantly tipped round the post by Fraser Forster and, from the resulting corner, Gordon Greer headed home to save a point for the play-off contenders.