Despite winning their last four, the hosts were behind after 15 minutes when Tutte collected a short corner on the left and his inswinging cross evaded everyone and nestled in the far corner.
The hosts responded when Danny Jackman's s free-kick was deflected behind, Deon Burton volleyed over from close range and Matt Fish fired over after a quick counter-attack.
However, Jackman lunged in on Tutte to receive a straight red card just after the half hour, and if Ryan Edwards had kept his cool when the resulting free-kick fell to him inside the box, the game could have been up.
The match became stretched after the break and the visitors took advantage, Tutte releasing Grant on the right and, after seeing his effort saved by Stuart Nelson, the winger drilled in the rebound.
Chris Whelpdale was hauled down by George Donnelly in the area and substitute Danny Kedwell converted the penalty to give the home side hope with 15 minutes to go, but Rochdale held on.
The result allowed second-placed Port Vale to close to within three points of the leaders with a 1-0 success over fellow high-flyers Bradford at Valley Parade,
The winner came against the run of play in the 37th minute when Vale's leading scorer Tom Pope intercepted a headed back-pass from Bantams skipper Gary Jones, beating goalkeeper Jon McLaughlin with a left-footed shot from just inside the penalty area to end the hosts' perfect home record.
Fleetwood and Accrington moved within five points of the Gills thanks to 2-1 wins over 10-man Barnet and AFC Wimbledon respectively.
Barnet keeper Graham Stack saw red as his side gave up an early lead, David Ball's penalty and Tom Barkhuizen's 78th-minute winner turning things around after Jake Hyde had bundled in an early opener.
Peter Murphy and Padraig Amond were the men on target for Stanley, with Luke Moore's stoppage-time effort scant consolation for Wimbledon as Simon Bassey slipped to his first defeat as their caretaker manager.
Elsewhere in League Two, Matty Blair and Jason Walker struck early in the second half to end York's four-match winless streak with a 2-0 victory at Aldershot.
Dagenham & Redbridge clinched their first win of the campaign in emphatic style - 3-0 at home to second-bottom Wycombe thanks to goals from Dwight Gayle, Michael Spillane (penalty) and Billy Bingham.
Skipper Kelvin Langmead scored at both ends as Northampton fought out a thrilling 3-3 draw at Burton.
Tommy Wright maintained his unbeaten run as Chesterfield caretaker boss as his side drew 1-1 with Torquay at the Proact Stadium.
Bristol Rovers picked up their first victory of the season with a 2-1 win at Exeter, Rotherham were 3-1 winners against Oxford and Plymouth's clash with Southend ended 1-1.