The 19-year-old Sinner, who reached the quarter-finals on his Roland Garros debut last year, was 5-4 down in the fourth set when Herbert sent a backhand wide to give the world number 19 a lifeline.
Sinner barely got out of first gear in a one-sided opening set before Herbert started to put pressure on his first serve, rushing to the net to unsettle the Italian.
The plan worked to perfection as Sinner misfired and appeared to lose his cool. But his powerful groundstrokes eventually wore Herbert out, who missed out on an upset a year after forcing German Alexander to a five-setter – which he had lost, too – in Paris.
“When I saw the draw I knew it would be tough, also knowing that the crowd is for him but it’s good to have crowds," said Sinner on court Suzanne Lenglen.
“Being a match point down and winning… it’s a crazy sport but I’m happy it went my way."
Sinner next faces fellow Italian Gianluca Mager on his way to a potential fourth-round clash with 13-time champion Rafael Nadal, who ended his run last year.