Juventus twice came from behind on Saturday night, and overcame a red card, but finally capitulated in the 90th when Zielinski drove in a low shot after being set up by Yann Bisseck near the penalty area.
The Derby d'Italia defeat dropped Juve (46 points) one place to fifth behind Roma, who have a better goal difference and can move into third if they win at defending champions Napoli (49) on Sunday.
"Juventus deserve credit," Inter coach Cristian Chivu said.
"They caused us problems with their rotations and style of play. They are a team in good form and even with ten men they equalised and put us under pressure ... But in the end, the team's character came out and they brought it home with a great goal from Zielinski."
Juventus left back Andrea Cambiaso scored at both ends of the San Siro in the first half.
He inadvertently deflected Luis Henrique's cross past his goalkeeper Michele Di Gregorio in the 17th minute, then latched on to a right-wing cross from American midfielder Weston McKennie in the 26th.
Juve's chances were hampered when defender Pierre Kalulu was sent off for a second yellow card late in the first half.
Inter (61) regained the lead in the 76th when forward Pio Esposito headed in powerfully from Federico Dimarco's left-wing cross. Manuel Locatelli equalised from another McKennie cross in the 83rd to ensure a tense finish, but Zielinski ensured Inter notched a sixth straight win overall.
Elsewhere, Como's hopes of securing Champions League qualification were dealt a blow by relegation-threatened Fiorentina (21).
Nicolo Fagioli and Moise Kean scored in 2-1 win for the visitors.
Cesc Fabregas' side pulled one back on 77 minutes but were reduced to 10 men when Spain international Alvaro Morata was sent off in the 89th.
The day got worse for Como (41) as Atalanta won 2-0 at Lazio to move one point above in sixth place, thanks to a penalty from midfielder Ederson in the 41st minute and a second-half effort from Poland's Nicola Zalewski.
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