Lautaro Martínez and Yann Bisseck also scored for the Serie A leaders on Saturday to help them move six points clear of Napoli, who hosts relegation-threatened Empoli on Monday.
Cagliari were six points above the drop zone.
Inter coach Simone Inzaghi kept the squad rotation at a minimum despite Wednesday's crucial match at San Siro, where the Nerazzurri will have a 2-1 lead from the first leg last week.
And Inter got off to the ideal start in the 13th minute. Carlos Augusto chested down Hakan Çalhanoğlu's ball over the top towards Lautaro but instead Arnautovic pounced on it and smashed it into the roof of the net.
Lautaro did get his goal 13 minutes later when an intelligent pass from Arnautovic sent the Argentine World Cup winner racing clear through on goal and he dinked it over the onrushing Elia Caprile.
Inter gifted Cagliari a way back shortly after halftime as Roberto Piccoli was left completely unmarked to head in a goal for Cagliari.
But Bisseck headed in a corner seven minutes later.
Dusan Vlahovic provided two assists to help Juventus beat relegation-threatened Lecce 2-1 and strengthen their hopes of qualifying for the Champions League.
Juventus moved into third place, a point ahead of Atalanta, who host Bologna on Sunday in a direct battle for the top four.
Bologna are only a point below Atalanta and two above Lazio ahead of the capital derby against seventh-placed Roma — also on Sunday.
In Turin, it took Juventus less than 100 seconds to open the scoring.
Vlahovic controlled a through ball from Khéphren Thuram and weighed his pass perfectly to send Teun Koopmeiners into the left side of the area for an angled drive into the far bottom corner.
Lecce almost responded immediately but Nikola Krstovic's effort came off the right post.
Instead it was Juventus who doubled their lead in the 33rd with a delightful team goal. Thuram played a one-two with Kenan Yıldız and then passed to Vlahovic who touched it on for Yıldız to fire into the bottom right corner.
Lecce captain Federico Baschirotto headed into the bottom left corner three minutes from time but it proved just a consolation for the visitors.
Venezia boosted their hopes of avoiding the drop with a crucial 1-0 win over bottom club Monza, who seem ever more likely to be relegated.
Daniel Fila netted his first goal for Venezia, since joining the Italian club from Slavia Prague in February.
Fila was also sent off in stoppage time following a second yellow card.
Venezia moved level with Empoli, two points from safety.
Monza remained 11 points below 17th-placed Lecce, with just six rounds remaining.
