The former Inter Milan midfielder failed to get on the score sheet but pulled the strings for Jurgen Klopp’s side in the first leg of their Europa League last-16 tie.
Daniel Sturridge made the breakthrough from the penalty spot after Netherlands international Memphis Depay brought down Liverpool defender Nathaniel Clyne.
Firmino added a second for the hosts from close range to secure three points, while Coutinho was denied by the acrobatics of David De Gea.
The former Hoffenheim midfielder didn’t look to bask in the glory of his goal as Firmino heaped praise on his Brazilian compatriot after Thursday night’s win.
Firmino wrote on his Instagram page: “TopPlayer #ynwa”
The South American has scored eight goals in 2016 so far, including an effort in a 3-0 win against Manchester City last week.
Liverpool make the return trip to United for the second leg next Thursday.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news