Jones' sacrifice fly in the first inning put the Pirates ahead to stay.
After Starling Marte and Travis Snider had each driven in a run, Jones capped a fourth-inning outburst with a run-scoring double, handing Pittsburgh a 6-1 lead.
What turned out to be the decisive run, however, came in the seventh when Clint Barmes socked a solo homer for a 7-3 Pirates advantage.
Andrew McCutchen went 2-for-3 with a solo shot for the Pirates, who broke a season-worst seven-game losing streak.
Wandy Rodriguez (11-13) earned the victory, allowing three runs (one earned) on nine hits over six frames, walking one and striking out four.
The Bucs' bullpen nearly blew the lead, but Joel Hanrahan managed to notch his 35th save despite yielding two runs on three walks and a pair of hits in the ninth.
Jason Berken (0-1) was rocked in his Cubs debut, surrendering six runs (two earned) on eight hits over four innings.
Luis Valbuena and Alfonso Soriano each knocked in a pair of runs for Chicago.