The Blades never looked threatened by Rovers as they produced a classic away performance, while Barry Robson and Ryan Flynn both went close.
Jonathan Forte had the game's clearest opportunity in the 24th minute, playing a neat one-two with Dave Kitson, but stand-in skipper Owain fon Williams was equal to his effort.
The Blades took the lead minutes before the break, with Tony McMahon's ambitious free-kick taking a huge deflection off Taylor as he tried to clear and flying past fon Williams.
Both sides were cagey after the restart, with only on-loan Manchester United man Sean McGinty testing George Long for Rovers.
Flynn nearly sealed all three points for United, but his fiercely struck effort five minutes from time smashed against fon Williams' bar and away to safety.
Robson went even closer a minute later, but former Rochdale keeper fon Williams tipped his superb strike over the bar.