The 39-year-old Argentine began his Inter career 17 years ago, playing in a 1-0 Serie A win over Vicenza on August 27, 1995, after joining from Banfield, an unfashionable Buenos Aires club.
Since then he has won five Serie A titles, four Coppa Italias, one Champions League, one UEFA Cup and one Club World championship.
Zanetti's fellow Argentines Esteban Cambiasso and Rodrigo Palacio scored Inter's goals in the play-off round first leg.
Thirty matches were played on Thursday involving teams ranging from five-times European champions Liverpool to Neftchi Baku from Azerbaijan.
Neftchi achieved a 1-1 draw at home to APOEL Nicosia, who reached the quarter-finals of the Champions League last season.
Russian club Anzhi Makhachkala, who have splurged millions on players like Cameroon forward Samuel Eto'o and Dutch coach Guus Hiddink, beat AZ Alkmaar 1-0 with a second-half goal from Ivorian forward Lacina Traore.
The match was played in Moscow after UEFA prevented Anzhi from playing at their base in the violence-plagued Dagestan region for security reasons.
Norway's Tromso, based inside the Artic circle, produced a shock 3-2 win over Serbia's Partizan Belgrade.
Athletic Bilbao, beaten finalists last season, trounced HJK Helsinki 6-0. Aritz Aduriz and Markel Susaeta scored twice each for Athletic with Inigo Perez and Andoni Iraola sharing the others.
- Champions League
- Neftchi Baku
- Rodrigo Palacio
- Esteban Cambiasso