Alex Revell squandered an early opportunity to put the npower League Two side ahead before Andy Webster flashed a header over from close range after 30 minutes.
But Hearts took the lead two minutes into the second half when Sutton slotted home from five yards from Ryan McGowan's pin-point cross.
Revell headed just wide and recent signing Kieran Agard was denied by Jambos goalkeeper Jamie MacDonald late on.
- Andy Webster
- Alex Revell