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With Liverpool 2-0 up at Old Trafford, Salah netted his first on 38 minutes to equal Drogba's tally before overtaking it with almost the final kick of the first half to put his side 4-0 ahead.
He added a fifth on 50 minutes and, in doing so, became the first visiting player to score a hat trick at Old Trafford in Premier League history.
Salah scored one goal in 13 appearances while at Chelsea in 2014, but the rest have come since he returned to English football to sign for Liverpool in 2017.
He won the Premier League Golden Boot with a single-season record of 32 in his first full campaign in the league in 2017-18 and was the joint leading marksman 12 months later.
Salah helped Liverpool win their first Premier League title in 2019-20 when they ended a run of 30 years without being crowned champions of England.
Drogba, who enjoyed two spells at Chelsea from 2004 to 2012 and 2014-15, won the Golden Boot on two occasions (2006-07 and 2009-10) and was a Premier League champion four times.
Salah, 29, has some way to go catch former Blackburn Rovers and Newcastle United forward Alan Shearer, who is the Premier League's all-time leading marksman on 260, while ex-Manchester City frontman Sergio Aguero is the top-scoring non-European with 184.