After a goalless first half, Burrow connected with 55th minute cross from trialist Jake Caprice to plant a firm header past James Ashdown.
Former Blackpool youngster Caprice, on as a replacement for a limping Tom Miller, had a hand in the second as well as after he forced Ashdown into a decent save, the resultant corner was cleared as far as skipper Alan Power who fired home an excellent second from 20 yards.
City: Paul Farman, Tom Miller, Sean Newton, Nat Brown, Tony Diagne, Arnaud Mendy, Todd Jordan, Alan Power, Conner Robinson, Jordan Burrow, Ben Tomlinson.
Subs: Luke Chambers (for Farman 62), Hamza Bencherif (for Brown 46), Jake Caprice (for Miller 35), Alex Simmons (for Robinson 72), Kegan Everington (for Jordan 72), Jordan Harper (for Tomlinson 85).
Doncaster (1st half): Ben Williams, Mitchell Lund, Luke McCullough, Peter Clarke, James Husband, Paul Keegan, James Bailey, James Coppinger, Lewis Ferguson, Harry Forrester, Theo Robinson.
Doncaster (2nd half): Jamie Ashdown, Harry Middleton, Ben Askins, Liam Wakefield, Cedric Evina, Dean Furman, Marc De Val Fernandez, Elliot Kebbie, Richie Wellens, Nathan Tyson, Alex Peterson.
Referee: Steve Ross
Assistants: Duncan Robertson & Jack Forder
Attendance: 1,144 (358 visiting supporters)