Eagles will face Manchester Magic, who finished top of Division Two North, in the last four.
They led by just three points going into the final quarter of a nip and tuck contest against the Bears.
Doncaster’s play-off experience then showed as they managed to pull away for the win.
Robert Marsden led the scoring with 16 points, ably supported by Eddy Brownell (11), George Colwill (11), Mathew Martin (9) and Josh Metcalf (9).
Eagles, who finished third in the regular season, travel to Manchester on Saturday.