The Brazilian found the net with a fine solo goal just seconds after half-time to add to Steven Gerrard‘s first-half opener from the penalty spot.
Jose Enrique then finished a fine team move by stabbing in a third before Luis Suarez got himself on the score-sheet with a neat left-foot finish from the edge of the box.
Daniel Sturridge also got in on the act with 20 minutes remaining, scoring the Reds’ second penalty of the afternoon to wrap up an excellent afternoon’s work.
The result ends Liverpool’s five-game streak without a win, and moves it into seventh place in the Barclays Premier League table.
Brendan Rodgers opted to hand Coutinho his first start for Liverpool, while fellow January signing Sturridge returned to the team following injury. There were also places for Lucas Leiva and Daniel Agger.
The Reds kicked off aiming to end a run of consecutive two-goal defeats to West Brom and Zenit St Petersburg — and Suarez was first to threaten, drilling straight at Michel Vorm after engineering some space inside the box.
Stewart Downing then arrowed wide of the near post from the edge of the box as the Reds slowly set about getting a grip of proceedings.
Next, Gerrard cannoned a free-kick from inside the D into a wall of Swansea bodies before Suarez lofted the rebound into the Kop.
In the 23rd minute, Liverpool fashioned its best chance of the afternoon so far when Downing weaved into space on the right before delivering a dangerous ball into the six-yard box, which Sturridge headed narrowly over.
An even better opportunity was to follow moments later when Suarez’s delicious outside-of-the-foot pass sent Sturridge scampering into the area. The striker attempted to negotiate his way beyond a clutch of Swansea defenders before seemingly being tripped by Kyle Bartley. Referee Howard Webb, however, waved play on as the ball fell for Coutinho, but he could only steer wide under pressure.
Liverpool’s domination was growing and a deserved goal finally arrived 11 minutes before half-time.
Suarez collected a Gerrard corner at the back post and attempted to work his way past Kemy Agustien, but was sent crashing to the turf by the Swansea No. 26, leaving Webb with little option but to point to the spot.
Gerrard may have missed a penalty in the last home game against West Brom, but there was never any doubt the skipper would step up once again — and this time he dispatched the ball into the bottom corner with trademark aplomb.
There was no sign of the Reds taking their foot off the pedal either, and only Bartley’s header off the goal-line denied Sturridge from making it 2-0 after he’d dinked the ball over Vorm.
The hosts were certainly worthy of more than a one-goal lead — and they did duly double their advantage just seconds after the break with an effort to savor.
Coutinho collected the ball on the left wing and charged at the Swansea defence before beating Vorm with a low effort from the edge of the box to illuminate Anfield.
The Brazilian was enjoying an impressive full debut and he almost claimed a second with an audacious back-heel that squirmed just past the post.
A third did arrive, though, after 50 minutes as Coutinho, Suarez and Sturridge all combined neatly before the ball was worked to Enrique on the edge of the box.
He played a pass into Sturridge, who picked out his run into the danger zone with a low cross and the Spaniard found the net with a close-range prod.
Swansea was visibly shell-shocked — and worse was to come for the Welsh side as Suarez bamboozled two defenders on the edge of the box before clipping into the bottom corner via his left foot from 18 yards. It was goal number 23 of the season so far for the 26-year-old.
Soon after, Coutinho was withdrawn from the action on the hour mark having earned himself a standing ovation as Jordan Henderson took his place.
However, the chances continued to come for the hosts with Sturridge twice thwarted by fine Vorm saves — but the No. 15 was not to be denied after 71 minutes.
The striker was afforded the chance to score from the spot after Wayne Routledge handled Enrique’s cross and made no mistake with a powerful finish after Gerrard generously stepped aside from penalty-taking duties.
The Reds could have added to their tally in the closing stages, but Rodgers will be hoping his side have saved some more goals for Thursday night when they attempt to overturn the two-goal deficit against Zenit St. Petersburg in front of the Kop.
Liverpoolfc.com journalists’ man of the match: Daniel Sturridge