Star striker Mohamed Salah returned to the squad as a second-half substitute from an injury that had restrained him to only play 46 minutes for Liverpool since his early retreat from the Africa Cup of Nations in January.
"Sparta took risks, we punished them for that," said Liverpool manager Jurgen Klopp.
"We scored wonderful goals and we had really good moments in the game."
Alexis Mac Allister opened the score against the reigning Czech champions and top flight leaders with a sixth minute penalty after being brought down by Asger Sorensen in the box.
Nunez added a second with a long-range shot from outside the box over diving Sparta goalkeeper Peter Vindahl in the 25th minute.
The Uruguay striker made it 3-0 in first half stoppage time with a low blast from just inside the box to the far post.
"He has the most important attitude a striker needs to have in these chances," said Klopp.
"He's just a constant threat and he gives us some spaces and areas and options."
Conor Bradley came on as a half-time substitute and the game turned sour for him 40 seconds later as he slammed Sparta winger Veljko Birmancevic's cross into his own net.
Luis Diaz, however, made it 4-1 on 53 minutes, beating Vindahl with a deflected shot and Dominik Szoboszlai, also returning from injury as a second-half substitute, rounded off the score in the stoppage time.
'Happy about the result'
Salah found the net six minutes from time but his goal was disallowed as VAR caught him offside.
Liverpool have all but secured a place in the March 15 draw for the quarter-finals, which will also reveal their potential semi-final opponents.
The Reds controlled much of the game while Sparta relied largely on breaks.
The hosts came close several times but goalkeeper Caoimhin Kelleher stopped unmarked Lukas Haraslin's close-range effort with his fingertips and Birmancevic then missed a gaping net.
"The result's much better than the performance," said Klopp.
"Our protection was not good and so they could create these counter-attacks. We lost a few unnecessary balls which is then difficult to protect."
"We made mistakes, that's human, but we fought really hard and I saw in the dressing room afterwards the boys felt the intensity of the game and were happy about the result."
Klopp made three changes from Saturday's Premier League game against Nottingham Forest.
Captain Virgil van Dijk only came on as a second-half substitute with Andrew Robertson wearing the band on his Europa League debut.
Liverpool are eyeing a remarkable quadruple in Klopp's final season as manager at Anfield as they have won the League Cup and are through to the last eight of the FA Cup.
They also top the Premier League, one point ahead of Manchester City whom they face on Sunday.
