The Canaries took the lead through a Zak Whitbread header in the 27th minute.
A free-kick from David Fox caught out goalkeeper Brad Guzan and Whitbread was left with an easy header from 12 yards.
But Barmby arrived at the near-post to level from close range midway through the second half.
The visitors are now just one point clear in second, while Hull kept alive their slim hopes of snatching a play-off place.
Source: PA
Source: PA