A contest between recent a champion and a new additions to the elite was a serving of the English top flight at its most exhilarating, although Liverpool boss Jurgen Klopp may take issue with his team's defending.
Ethan Pinnock and Diogo Jota traded early goals at Brentford Community Stadium, before Salah and Vitaly Janelt did likewise early in the second half.
Curtis Jones then scored with a 25-yard blast, helped by a slight deflection, but Yoane Wissa had the final say, his clever finish in the 82nd minute meaning Liverpool goes just one point clear at the top of the table rather than three.
Salah looked sure to get his landmark goal in Liverpool colours in the seventh minute when he poked Jota's pass under goalkeeper David Raya, only for Kristoffer Ajer to produce a sublime goal-line clearance.
Ivan Toney headed off target and Frank Onyeka fired over while stretching as the hosts challenged their illustrious visitors, before Bryan Mbeumo became the second player denied by a clearance off the line, Joel Matip getting back in time.
Those were warning shots, but Liverpool was hit by a sting from the Bees when Pinnock put Brentford ahead in the 27th minute. The defender got on the end of a clever free-kick routine, with Toney's deft flick allowing Pinnock to slide in at the far post.
Liverpool was level four minutes later, Jota powerfully heading a first-time cross from the right by Jordan Henderson past Raya. Jones rattled the left post from 22 yards before the break, Raya saving brilliantly as Jota reacted to the rebound.
Salah made no mistake in the 54th minute, however, slotting in Fabinho's excellent pass over the top. He was declared offside initially, but replays showed he was not beyond the last defender.
Brentford was not done, and when Pontus Jansson smashed a shot against the Liverpool bar, Janelt was on hand to head in at the second attempt, beating Virgil van Dijk to the ball.
Jones silenced the home crowd with a thumping drive that took a flick off Ajer, Liverpool going 3-2 ahead in the 67th minute.
Salah floated the ball a foot over the bar when he should have fired in Liverpool's fourth, and it proved costly when Wissa won the race for the ball in a scramble and chipped over Alisson.
Toney thought he had won it in the 87th minute when his sizzling strike beat Alisson, but he had stepped a yard offside.