The lower league side almost took the lead after just 12 minutes when Robbie Blake's shot looked goal bound, but keeper Mark Bunn was equal to it.
The Canaries came close twice in the space of a minute soon after, first through Harry Kane, who saw his shot blocked after Doncaster's Garry Woods fumbled. Then Woods had to be alert to tip Marc Tierney's speculative shot onto the crossbar.
The hosts did take the lead after 25 minutes when Grant Holt did well before finding Alexander Tettey, who took a touch before firing his shot in off the post from 22 yards.
The lead was almost doubled on the stroke of half-time when Elliott Ward's close-range header was cleared off the line by Jamie McCombe.
Woods kept the score at 1-0 seven minutes after the restart with a fine double save, first from Holt and then from Simeon Jackson.
Doncaster almost got a dramatic late equaliser when James Husband's low cross found David Cotterill, but the midfielder could not connect properly from just one yard out.
The Premier League side held out for the rest of the match, safely securing their place in the pot for round four.