Firmino, 23, joined Liverpool from Bundesliga club Hoffenheim in a £28 million deal this summer, making his Premier League debut off the bench in Sunday's 1-0 win at Stoke.
The Brazil international scored three goals for Liverpool on Monday at the club's Melwood training ground, while Lazar Markovic and fellow new signing Danny Ings were also on target.
The match against TNS was an opportunity for those players who had not started against Stoke to build up fitness for the new campaign, with Adam Bogdan, Emre Can, Lucas Leiva, Alberto Moreno, Divock Origi, Jordan Rossiter, Mamadou Sakho and Kolo Toure also playing.
Firmino made his Liverpool bow from the substitutes bench at the Britannia Stadium on Sunday, playing the final 12 minutes in place of Jordan Ibe as fellow Brazilian Philippe Coutinho scored a late winner to avenge last season's 6-1 humiliation on the same ground.
