Cagliari, who won their only Serie A title in 1970, will follow Parma and Cesena into Serie B after an unhappy season during which they employed three different coaches.
Zdenek Zeman was in charge until December when he was replaced by Gianfranco Zola, who was sacked in March. Zeman returned but quit one month later and Gianluca Festa took his place.
Cagliari, 18th with 28 points, never recovered after Franco Vazquez curled the ball in after nine minutes and they were jeered off at the end.
Atalanta, the only team Cagliari could have caught, were able to celebrate survival despite being crushed 4-1 at home by fifth-placed??Genoa.
Chile forward Mauricio Pinilla gave Atalanta the lead with a penalty but Leonardo Pavoletti headed Genoa level on the half hour.
Andrea Bertolacci put Genoa ahead with a superb solo effort in the 57th minute, Iago Falque fired in another four minutes later and scored the fourth with another curling effort.
Domenico Berardi scored a hat-trick against AC Milan for the second season in a row to give Sassuolo a 3-2 win over Filippo Inzaghi’s team, who had Giacomo Bonaventura and Suso sent off in the second half.
Angry Milan, level on 46 points with 10th-placed Palermo after a dismal season, claimed Berardi’s first goal did not cross the line after goalkeeper Diego Lopez failed to hold his shot and clawed the ball away.
Roma??stayed second, on course for a place in the Champions League group stage, after coming from behind to beat??Udinese??2-1.
Roma have 67 points, 16 behind champions Juventus, who beat Inter Milan 2-1 on Saturday, and one more than third-placed Lazio who won 1-0 at Sampdoria. Napoli, fourth with 60 points, host Cesena on Monday.
Stipe Perica put Udinese ahead from their first real chance in the 19th minute, the 19-year-old’s first goal since arriving on loan from Chelsea in January.
Radja Nainggolan took full advantage of being left unmarked as he met Francesco Totti’s inviting cross with an emphatic half-volley on the stroke of halftime and Vasilis Torosidis scored the winner midway through the second half.
Second place earns a spot in the Champions League group stage next season and the team finishing third will qualify for the final playoff round.