United trailed for 50 minutes at the Stadium of Light after the Black Cats, who are still without a Barclays Premier League win this season, took a shock lead.
However, teenager Adnan Januzaj capped his full debut for the club with a stunning double to end a run of two successive league defeats.
Jones told MUTV: "Everyone dug in, it was a great performance and a massive three points.
"We were put on the back foot right from the first minute, and perhaps we didn't get out of the traps as well as we should have done.
"They were fired up, and perhaps we didn't expect it quite as much as we should have in the first 10 minutes. But once we got to grips with the game, I thought we played some terrific stuff.
"That was the sign of champions out there. When your backs are against the wall and everyone's raising questions, a performance like that puts the record straight."
Jones in particular endured a difficult start to the game when his attempted clearance from Emanuele Giaccherini's fifth-minute cross hit team-mate Nemanja Vidic and dropped nicely for Craig Gardner to drill home the opening goal.
It might have been 2-0 within four minutes after Jones was robbed by Gardner, who ran in on goal, but then dithered when a shot or a cross to the supporting Jozy Altidore might have put United in deeper trouble, allowing the defender to make a vital interception.
However, it took a superb save by keeper David de Gea to keep out Giaccherini's 34th-minute header and the Black Cats headed for the dressing room at half-time good value for their lead.