- Lautaro Martínez (62', 77', 85' Pen, 89')
The Argentine forward is the first player to score four goals off the bench in the last 30 seasons in Serie A, since three points for a win were introduced in 1994-95.
The hosts had kept Inter at bay for the first hour, but fell apart after Martinez came on nine minutes after the break and bagged four goals in 27 minutes, including one from the penalty spot.
Inter had dropped three points below AC Milan in the table after their rivals won 2-0 against Lazio earlier on Saturday, but returned to the summit with 18 points from seven matches. Salernitana are 19th with three points.
Inter captain Martinez has now scored nine goals in seven Serie A matches and holds a clear lead as the league's top scorer.
"I scored four goals but the important thing is that Inter won the game," he told Sky Sport Italia, adding that turning things around after the 2-1 loss to Sassuolo on Wednesday had been a priority.
"It was crucial for us. We lost a game at San Siro that I don't think we deserved but we conceded two goals from our errors, so we had to win today."
While Inter dominated the match's early stages, Salernitana gradually found their footing and managed to control the tempo in a goalless first half.
Three minutes later, Mateusz Legowski's apparent equaliser was disallowed due to an early offside run, eliciting boos from the home fans.
Martinez got his second goal after 77 minutes when he confidently finished a pass from Nicolò Barella, who set him up from 12 yards out, with a first-time strike.
Martinez completed his hat-trick from the penalty spot with five minutes remaining, after a foul by defender Matteo Lovato, who had pulled Thuram's shirt.
He then concluded the evening by converting Carlos Augusto's cross, firing from 10 yards out, a minute before stoppage time.
Inter manager Simone Inzaghi had high praise for Martinez.
"It is an excellent start but in my two years here he was always consistent and tonight was the time to have a little rest," he said, referring to Martinez's start on the bench.
"He is our leader, the captain, he must continue like this."
- Rosario Abisso