Darwin Nunez had the ball in the net twice in the first half, only for both of his strikes to be flagged for offside.
However, the Uruguayan's constant involvement in Liverpool's attacking play paid off in the 39th minute, turning provider to slot through Salah, who fired past Mark Flekken to give the Reds a vital lead.
Salah would have his second after the break, converting with a simple header after Kostas Tsimikas did brilliantly to keep the ball in play to cross.
Jota would seal the points in the 74th minute, dancing past a couple of Brentford defenders before curling his effort into the far corner.
Here are the player ratings from the match.
ALISSON (GK): 8
Massive save to deny Bryan Mbeumo in the first half, who went one-on-one with the Brazilian shot-stopper after a rapid breakaway from Brentford. Kept the momentum in Liverpool's favour. Deserved clean sheet.
TRENT ALEXANDER-ARNOLD (RB): 7
Ambitious with his passing and involved in the goal. Defensively sound as well.
JOEL MATIP (CB): 7
Couldn't contain his disbelief when Paul Tierney blew for a completely fair challenge, and was subsequently booked. Defensively sound otherwise.
VIRGIL VAN DIJK (CB): 8
Could have easily scored the opener but saw his header saved brilliantly by Flekken in the first half. Composed at the back, brilliant defending after the break to prevent Mbeumo from getting a shot away in the box.
KOSTAS TSIMIKAS (LB): 8
Did brilliantly to keep the ball in play for the second goal, then set up Jota for the third. Also improved defensively after a couple of rocky performances.
WATARU ENDO (CM): 7
Simple but effective in the 'six'. Tense moment in the second half when VAR spent what felt like an age checking over his challenge for a possible red card, but the claims were eventually dismissed. Perhaps lucky to stay on the pitch, but otherwise solid.
DOMINIK SZOBOSZLAI (CM): 7
Nice link-up play with his teammates and aided Liverpool's progression in the final third.
CODY GAKPO (CM): 7
Link between midfield and attack. Took the first shot of the game - a beautifully-struck effort from outside the box that would have caused Flekken some problems if it wasn't straight down his throat.
MOHAMED SALAH (RW): 8
Slotted past Flekken with aplomb to give Liverpool a crucial breakthrough goal, then doubled his tally for the afternoon with a simple header. That's 200 goals in English football. An all-time great.
DARWIN NUNEZ (ST): 8
Great play to slot Salah through for the opener, finally getting his goal contribution after having two goals disallowed for offside - one by the most minimal of margins. Constant involvement up top.
DIOGO JOTA (LW): 8
Put a lot of pressure on the Brentford defenders and looked sharp throughout. Bagged a brilliant goal in the second half, firing into the bottom right corner to seal the points.
Luis Diaz (Diogo Jota, 83'): 6
A feel-good moment as Diaz made his Anfield return.
Harvey Elliott (Cody Gakpo, 83'): 6
Helped see out the contest.
Jarell Quansah (Trent Alexander-Arnold, 90'): N/A
A couple more minutes for the young defender.
James McConnell (Dominik Szoboszlai, 90+4'): N/A
First taste of Premier League football for the 19-year-old.