The hosts, who started the game four points clear of the teams in the relegation zone, wasted little time in testing the Boro defence as Billy Sharp played in Richard Wellens, but his shot from a tight angle was blocked.
As the game continued at a furious pace, Doncaster went on probing the visitors' defence with both Paul Keegan and James Coppinger having chances.
Doncaster also went close to breaking the deadlock when Sharp found Chris Brown, but his shot was saved by Given.
Paul Dickov's side continued their bright start to the game and could have been ahead on 28 minutes had Wellens kept his shot down.
As the interval approached, Doncaster again went close to scoring with Coppinger getting onto Brown's headed pass, only to see his effort blocked.