Friday's win moved Inter 10 points clear of second-placed Juventus, who play at Verona on Saturday.
Marcos Thuram opened the scoring with a fine side-foot finish after 16 minutes and Lautaro Martinez doubled the lead three minutes later with a shot from outside the box.
Denzel Dumfries made it 3-0 five minutes before half-time when he tapped in from close range after the goalkeeper spilled a shot from Nicolò Barella.
Austrian centre forward Marko Arnautovic rounded off the rout in the final minute.
It was a sobering start for Fabio Liverani, who was appointed Salernitana coach last Monday with the task of keeping the club up.
His new side remained rooted to the bottom of the table, five points adrift of Cagliari and with just two wins all season.
Earlier, Duván Zapata scored his third goal in four games as Torino's super start to 2024 continued after comfortably beating Lecce 2-0.
Torino extended their unbeaten run to six games and rose one place to ninth in the standings.
Raoul Bellanova got the opener with a rifled shot from just inside the box five minutes into the second half. It was the midfielder's first goal of the season.
Lecce centre half Marin Pongracic was sent off with 19 minutes remaining, and nine minutes later Zapata scored with a textbook header from a corner kick.
Lecce are 13th after only one win in their last nine league games.
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