The Reds were smarting from Saturday's 2-2 draw with Brighton that saw them throw away a two-goal cushion and drop points in front of their fans.
Meanwhile, Atletico Madrid were looking to enact revenge for their 3-2 loss to Liverpool on their own turf a fortnight ago.
Things, however, got off to a poor start for the Spaniards, as Jurgen Klopp's men controlled proceedings from the first blast of the whistle.
Liverpool deservedly went ahead when an unmarked Diogo Jota evaded the Atletico defense to head home a precise cross by Trent Alexander-Arnold in the 13th minute.
Their lead was doubled just eight minutes later, and it was Alexander-Arnold who was once again the provider. The England international hit a low ball across Jan Oblak's goal, and Mane converted it from close range.
Atletico Madrid's job was made more difficult when Felipe was sent off for a cynical challenge on Mane in the 36th minute. However, replays showed the tackle was innocuous, and the Brazilian was unlucky to receive his marching orders.
The second half began with Jota seemingly scoring Liverpool's third goal just three minutes after the restart. However, the goal was chalked off for the most marginal of offside decisions.
Luis Suarez was heavily booed with every touch of the ball at his former stomping ground, and the Uruguay international barely had a sniff of Alisson's goal. He scored a deflected shot in the 57th minute that was disallowed by the VAR due to offside.
Despite some late pressure, the hosts were unable to add to their lead. The victory saw Liverpool stretch their unbeaten run to 25 matches in all competitions, while Atletico Madrid dropped to the third spot in Group B.
Here are five talking points from the game.
#5 Liverpool secure progress to the knockout rounds with ease
When the draws for the UEFA Champions League group stage were made, many pointed to Group B as potentially the trickiest group to navigate.
On the one hand, there were two bonafide elite teams in Liverpool and Atletico Madrid. They were among the favorites to go all the way in the tournament.
Porto and AC Milan rounded up the teams in the group. The latter might be participating in the tournament for the first time in seven years, but their recent ascent meant they could not be written off. On their day, Porto have the pedigree and quality to upset some of the biggest clubs in the world.
In light of this, qualification in the group was expected to be a keenly-contested affair, but Liverpool have thrown all pre-campaign permutations out the window.
The Reds have easily secured qualification to the next round with maximum points garnered from their four matches so far.
They have scored 13 goals and conceded five, with home and away victories over Atletico Madrid guaranteeing them top spot more easily than was expected.
#4 Atletico Madrid still have a fight on their hands to qualify
While Liverpool have already booked their spot in the last 16 as group winners, the remaining qualification spot in the group is still up for grabs.
Porto's draw with AC Milan earlier in the day meant the Dragons climbed into second spot, while the Rossoneri secured their first point of the campaign.
Despite being at the bottom of the standings with a sole point, AC Milan are still in with a shot at securing qualification.
Atletico Madrid were the second favorites to progress from the group after Liverpool. However, Diego Simeone's men fluffed their lines in their double-header with the Merseyside giants.
They still have their destiny in their hands, but Porto and AC Milan are likely to give their all to also secure qualification.