Salah's goal was his 20th this season in all competitions and put him two clear of Tottenham Hotspur's Harry Kane at the top of the Premier League scoring charts on 14.
With less than 11 minutes played, Nathan Ake was booked for tugging Firmino down on the edge of the box and Coutinho saw a sweetly-struck free-kick hit the inside of the right-hand upright and bounce away from goal.
Sadio Mane was given a rest by Liverpool manager Juergen Klopp but the other three members of the so-called "Fab Four" attacking unit - Salah, Coutinho and Firmino - played and scored.
Oxlade-Chamberlain quickly interjected to help Coutinho out, declaring that the No.10 is "a Liverpool player right now" and hailing him for "doing really well" in spite of constant links to Barcelona in the press. "We had a few games like the West Brom game - not performance-wise, but result-wise".
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Coutinho was also asked about whether his first-half goal was his best ever strike for Liverpool, but focused more on the benefit of getting back to winning ways after successive draws.
Begovic then denied Liverpool what would have been a stunning goal when Coutinho flicked a ball over the defence for Salah, who shot straight at the goalkeeper.
Alex Oxlade-Chamberlain nearly added to the score with a shot which cannoned back off a post in the second half before Firmino glanced home a header from another excellent Coutinho cross to confirm victory.
"I am really pleased about everything", Klopp gushed in his post-match interview.
"I wasn't expecting him to score that many goals, especially as a winger". "We were flat today for whatever reason, we never got going. But we have to be consistently good because we are Liverpool - that's what we have to learn".