AFC Bournemouth took the lead on nine minutes when Jefferson Lerma squared for Callum Wilson to tap in.
Liverpool equalised 16 minutes later. Mane won possession from Cherries substitute Jack Simpson to find Salah, who fired into the bottom corner.
Mane put Liverpool in front on 33 minutes, receiving Virgil van Dijk's through-ball before slotting past Aaron Ramsdale.
After the break, Ryan Fraser's effort was brilliantly cleared off the line by James Milner before Mane curled against a post.
A return to winning ways sends Liverpool 25 points clear of Manchester City, and only three wins from a first Premier League title.
AFC Bournemouth remain in the bottom three on 27 points.
Liverpool: 16 Mar v Everton (A)
AFC Bournemouth: 14 Mar v Crystal Palace (H)