City were good value for their point after producing a tremendous hard-working display which brought to an end Doncaster's four-match winning streak.
Both sides produced some good football despite their respective positions in the basement of the Championship, and if they continue in this vein they should pull clear of the relegation battle.
Doncaster found it hard to get their slick passing game going as Norwich closed them down quickly right from the start.
The Canaries were also eager to attack themselves and Lee Croft had a low shot deflected just past the post in the tenth minute.
Doncaster gradually began to get their act together with skipper Brian Stock playing well in midfield and a low shot from Matt Mills was cleared off the line by Grounds following good work by Richie Wellens in the 22nd minute.
Rovers snatched the lead a minute later when Sam Hird set up Irish striker Paul Heffernan to nod home from close range following a corner.
Norwich's response was immediate and Darel Russell headed just wide following a free-kick from Sammy Clingan in the 26th minute. Clingan almost caught home keeper Neil Sullivan unawares eight minutes later with a quickly-taken set-piece, but the ball sailed narrowly wide.
The end-to-end battle continued in the second half and Norwich were finally rewarded in the 61st minute when Grounds headed firmly into the centre of the net in a crowded penalty area after latching on to a right-wing cross from Croft.
Doncaster had a couple of penalty appeals turned down and they looked to have a fair case when James Coppinger was pushed in the back by Gary Doherty in the 65th minute.
John Otsemobor blasted a good chance wide for Norwich four minutes later but it was Rovers who finished the stronger side and went close several times to snatching a winner.
Visiting keeper David Marshall denied Coppinger when he tipped a fierce shot over the bar in the 70th minute and Mills missed a good chance when he volleyed wide three minutes later.
Substitute Gareth Taylor went close in the 77th minute for Doncaster but City held firm to earn a deserved point.