Jones struck either side of the break to hand the Black Cats a first home victory of the season, although they had to come from behind and rode their luck as Rovers, without Aston Villa target Stephen Warnock in the 18, threatened to leave with at least a point.
Blackburn took the lead through Gael Givet's 21st-minute strike, but despite the visitors enjoying the better of the first half, Jones levelled after 32 minutes and then headed his side in front nine minutes after the break.
However, Sam Allardyce's men had two goals disallowed and saw £6million signing Nikola Kalinic pass up four glorious opportunities, one of them after Christopher Samba had hit the bar 15 minutes from time.
That will have mattered little to the bulk of a crowd of 37,106, who left happy with six points from their side's first three games and hopeful of a much more comfortable season than the last two.
The visitors had already had a Jason Roberts goal correctly ruled out for a foul by Samba on goalkeeper Marton Fulop when former Black Cat El-Hadji Diouf dispatched a volley which looked to be heading for the top corner before Fulop threw out a hand to turn it away.
The respite, however, was short-lived as Rovers took the lead from the resulting corner when Lee Cattermole on the line could only clear Morten Gamst Pedersen's cross as far as Givet, who gleefully thumped a left-foot drive through a crowded penalty area and into the back of the net with the help of a slight deflection.
Sunderland looked vulnerable every time the visitors were presented with a set-piece with Samba proving a real nuisance in the box.
However, they were back on terms 13 minutes before the break after staging one of what proved to be a handful of cohesive attacks during the opening 45 minutes.
Cattermole's clearing header was picked up by Steed Malbranque, who lifted the ball over the top for Jones to leave Samba and Ryan Nelsen in his wake before rounding keeper Paul Robinson and slotting into the empty net.
Kalinic, on for the injured Franco di Santo, found himself in space 10 yards out, but scuffed his left-foot shot horribly.
Ryan Nelsen, who had just returned from injury, failed to reappear for the second half and his place was taken by Martin Olsson, although the change did not seem to hamper his team.
Indeed, Rovers might have gone ahead once again within five minutes when Fulop spilled Diouf's cross under pressure and Kalinic lifted the rebound high over the bar from four yards.
Blackburn were made to pay with 54 minutes gone when Cattermole broke down the left and crossed for Jones to power home a close-range header.
But the visitors thought they were back into it within seconds when, after Fulop had punched a Pedersen free-kick out to Steve Nzonzi, the midfielder looped a header into the empty net.
However, the flag had gone up for offside against Samba, and Allardyce's furious reaction said everything about his view on the decision.
Rovers' misery increased as the game entered its latter stages, with luck deserting them once again.
Samba looked odds-on to score after the ball was played back across goal and he got his head to it, but his effort came back off the underside of the crossbar and Kalinic was once again unable to stab it home in the midst of a melee.
Olsson ripped a shot into the side-netting seconds later with Sunderland living dangerously, but it was Kalinic who might have snatched a point deep into injury-time.
Fulop again flapped at a high ball and the ball fell to him 15 yards out, but he dragged his shot hopelessly wide of the target.