Jota headed in the opener three minutes into the second half following a corner but Forest drew level when former Liverpool fullback Neco Williams ran on to Morgan Gibbs-White's pass to send a deflected shot past Alisson.
Jota restored Liverpool's lead with a sharp finish on the turn in the 55th minute but Liverpool were pegged back again when Virgil Van Dijk's clearing header fell to an unmarked Gibbs-White, who fired an unstoppable shot into the corner.
Liverpool were back on top three minutes later when Trent Alexander-Arnold picked out Salah with a low free kick and the Egyptian striker held off Remo Freuler before side-footing past Keylor Navas.
Forest came agonisingly close to snatching a point when Taiwo Awoniyi sent an acrobatic effort over the bar and substitute Brennan Johnson hit the crossbar with a delicate chip with Alisson beaten, but Liverpool clung on to take all three points.
Juergen Klopp's men moved on to 50 points, one behind sixth-placed Aston Villa, while the defeat saw Forest drop to second-bottom in the table.
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