Chances were scarce in the opening 45 minutes, as Federico Macheda and James Coppinger both tried to break the deadlock for the home side, while Ben Mee and Sam Vokes saw efforts miss the target for Burnley.
However, the visitors were awarded a penalty in the final minute of the first half after Vokes was brought down by Jones, allowing the striker to put Burnley ahead from the spot.
Jones almost handed Burnley a second after the break, nearly turning a Kieran Trippier cross into his own net, but it went just wide for a corner.
Doncaster's first chance of the second half fell to Macheda, but the striker smashed his strike from 10 yards fractionally high, allowing Burnley to go on the counter-attack, quickly creating a chance for Danny Ings at the other end, which Jones managed to clear.
The home side almost drew level in the 81st minute, with Jones continuing to look to make amends for giving away the earlier penalty, but his header from a Coppinger corner went inches high of the crossbar.
Ross Turnbull deflected a Scott Arfield shot wide a corner late on, which Trippier curled into the box, resulting in a scramble on the goal line, with the final touch coming off Jones for an own goal just before full time.