Paul Lambert's side started slowly in Tuesday night's Championship encounter, but stole into a first-half lead through leading goalscorer Grant Holt.
However, Canaries defender Drury headed John Oster's second-half corner into his own net and Rovers held on to end a run of four straight defeats.
Rovers chairman John Ryan had wanted the game called off because of his side's crippling injury list and with just one win in their previous nine league matches, his concerns were understandable.
But the visitors took the game to Norwich in the first half and after withstanding a storm in the second half they struck back for a deserved point.
Henri Lansbury, on loan from Arsenal and looking to score in his third straight game, tested Doncaster goalkeeper Neil Sullivan with an early volley.
But midfielder Frank Moussa, Doncaster's new loan signing, had a goalbound shot blocked by Norwich defender Elliott Ward and was the game's outstanding player for the first half an hour.
Doncaster, without 11 first-team players, were the better side during this period, but Norwich took a 31st-minute lead against the run of play with only their second real effort on goal.
Canaries defender Drury broke free down the left and when he sent the ball skidding low across the six-yard box, Holt showed sound technique to slam the ball home first time, high into the roof of the net for his 15th goal of the season.
Norwich looked more assured after their breakthrough and visiting defender Wayne Thomas did well to block Lansbury's 20-yard effort, while Holt steered a gilt-edged chance from another dangerous ball in from Drury over the crossbar just before half-time.
Aaron Wilbraham fired a 20-yard shot just wide for Norwich after the restart, but Moussa again troubled the home defence, first drilling in a thumping drive that Norwich goalkeeper John Ruddy held and then weaving his way into the penalty area before being bundled off the ball.
Holt volleyed wide following Lansbury's cross just before the hour-mark as Norwich looked to build on their lead and then Wilbraham twice went close with headers.
The first, from Russell Martin's cross, was brilliantly parried by Sullivan and the second, from David Fox's free-kick flew wide.
Andrew Crofts then dragged his shot wide, but just as Norwich appeared favourites to double their lead, Doncaster hit back against the run of play.
Rovers midfielder Oster sent over an inswinging corner and Drury inadvertently headed the ball into the opposite corner of the net in the 83rd minute.
Moussa then flashed a volley wide for the visitors, while Norwich responded when Zak Whitbread's header was blocked by Rovers defender Wayne Thomas in the closing stages.