Friday's win marked Empoli's fifth game in a row undefeated, the club's best start to a season in Serie A.
Anjorin made his full debut after a summer move from Chelsea and his neat pass through the defence set up Lorenzo Colombo after 33 minutes.
Sebastiano Esposito added a second three minutes into the second half.
Cagliari have not hit the net in their last three games. They have two points from five games.
In Verona, Torino went top of the league with a 3-2 win. The home side played most of the match with 10 men after Pawel Dawidowicz was sent off in the 21st minute.
Torino took the lead after 10 minutes when poor defending allowed Antonio Sanabria through on goal, and the Paraguayan striker made no mistake.
Grigoris Kastanos equalised two minutes later when he smacked a stunning drive into the roof off the net from 20 metres out.
The towering Duvan Zapata put Torino back in front with a trademark header 12 minutes before the break, and Scotland striker Che Adams sealed the result with a low drive from 25 metres out 12 minutes from the final whistle.
Daniel Mosquera pulled one back for Verona in stoppage time.
Torino are one point ahead of Udinese and two clear of Napoli, both of whom have played a game less.
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