Liverpool's front three were all on the scoresheet, with Salah becoming just the eighth player - and the first African - to reach such a landmark in the Premier League era.
Jurgen Klopp's side move to within one point of Manchester United in second, although the Red Devils do have two games in hand on their North-East rivals.
The Champions League semi-finalists were immediately onto the front foot and their early pressure told within seven minutes when Sadio Mane notched his 17th goal of the season.
Bournemouth's goalkeeper continued to be kept busy as he pushed away a free kick from Trent Alexander-Arnold, and comfortably held Salah's low 25-yard curler.
While the Cherries offered no attacking threat in the first half - lone striker Jermain Defoe had only three touches in the opening 45 minutes - they defended resiliently to reach the break only one goal down.
Nathan Ake had shackled Salah fairly well before the hour mark, but the PFA Player of the Year nominee began to knock on the door, shooting straight at Begovic after a impressive run from Alexander-Arnold.
Salah was aggrieved not to be awarded a penalty following a push in the back from Ake on 68 minutes, but was celebrating less than 60 seconds later after skilfully directing Alexander-Arnold's cross over Begovic.
Firmino wasted a golden chance on 75 minutes by trying to check back onto his left foot when through one on one, and Bournemouth could have set up a nervy finish had they taken one of their late opportunities.
Loris Karius was finally called into action on 82 minutes when he reacted well to save from Dan Gosling, and the German goalkeeper was beaten by a left-footed Callum Wilson shot which drifted wide of the post.
The icing on the cake for Liverpool came with Firmino's 90th-minute third, as the Brazilian squeezed a shot into the bottom corner from 15 yards out, following a good run and pass from Alex Oxlade-Chamberlain.
Liverpool will be hoping that Bournemouth, who remain 11th in the table but will still have ambitions to finish in the top half, can do them a favour when United visit the Vitality Stadium on Wednesday.
LIVERPOOL (4-3-3): Karius; Alexander-Arnold, Van Dijk, Lovren (Klavan 83'), Robertson; Henderson, Wijnaldum, Oxlade-Chamberlain; Mane (Milner, 77'), Firmino (Solanke, 90'), Salah
BOURNEMOUTH (3-5-1-1): Begovic; Francis, Cook, Ake; Fraser, Cook (Surman, 73'), Gosling, Ibe (Mousset, 58'), Daniels; King; Defoe (Wilson, 70')