Cole Stockton grabbed the only goal of the game just before the hour mark at Prenton Park as Rovers took advantage of a slip-up by Sheffield United.
The Blades lost 3-2 at home to bottom club Hartlepool, who picked up just their second win of the season.
Steve Howard put the visitors ahead only for Michael Doyle to level. But a Ritchie Humphreys strike and Howard's second put Hartlepool clear, and Harry Maguire's late effort was not enough for United.
Meanwhile, Doncaster are now level on points with their second-placed Yorkshire rivals after a 3-0 win at Preston. David Syers opened the scoring inside the first minute, with David Cotterill and a Billy Paynter penalty completing the scoring.
Brentford dropped to fourth after a goalless draw at Shrewsbury while Bournemouth were 3-0 victors over Crawley thanks to a Gary Alexander own goal and further strikes from Brett Pitman and Eunan O'Kane.
Portsmouth remain in the bottom four after a 2-1 home defeat against Yeovil, while Crewe won at Oldham by the same scoreline.
Coventry came from behind twice before beating MK Dons 3-2 at stadiumMK. Daniel Powell and Ryan Lowe netted for the Dons but Franck Moussa and Stephen Elliott pegged them back before Elliott also added the winner.
Elsewhere Moses Odubajo struck three minutes from time to help Leyton Orient beat Walsall 2-1 and two early goals were enough to give Carlisle a 2-1 win over Bury.
The games between Colchester and Swindon and Stevenage and Notts County were postponed due to waterlogged pitches.