Liverpool had far too much attacking power for Bournemouth to withstand as they moved back into the top four with a 4-0 win at the Vitality Stadium.
Jurgen Klopp's side were 3-0 up at half-time as they created a string of chances, with defender Joe Gomez getting forward to have the game's first effort only to see it drift wide.
The visitors continuing to make the running, with only Lewis Cook's smart intervention preventing a Coutinho pass from giving Mohamed Salah the chance to run in towards goal.
Nathan Ake was booked for bringing down Roberto Firmino just outside the area to give away an 11th-minute free kick that was clipped against the post by Philippe Coutinho with goalkeeper Asmir Begovic struggling.
Soon afterwards, Salah twisted away from Charlie Daniels inside the area but, from an angle, sent a bending effort high into the stand.
Former Liverpool player Jordon Ibe then fired an effort into the arms of Simon Mignolet after Jermain Defoe had helped the ball on to him.
But after 20 minutes Liverpool broke through in style as Coutinho set off on a run, beating two defenders inside the area and creating the space to fire a shot low into the corner.
Soon afterwards Salah tried another curler but again sent the effort too high, but Bournemouth were two down after 26 minutes when a corner was helped back across by Firmino and Dejan Lovren dived to head the visitors' second.
They suffered another setback when forward Josh King took a knock and was unable to continue, replacing him with Junior Stanislas.
But with 39 minutes gone they were within a whisker of getting back into the game as Defoe raced through and sent a low shot beyond Mignolet, only to see it hit the inside of the post, with Begovic then making a good stop from Alex Oxlade-Chamberlain a few seconds later.
With 43 minutes gone, Begovic made another fine save as Salah blasted a shot from close range after a brilliantly engineered one-two with Coutinho.
But a minute later Salah made it 3-0 to Liverpool, again turning away from Daniels and this time finding the far corner with a curling strike.
Bournemouth boss Eddie Howe made a half-time change, bringing on Ryan Fraser for Marc Pugh, but a misjudged header from Nathan Ake almost gave Liverpool a fourth soon after the restart only for Salah to fail to keep the ball under control.
Firmino nearly got on the end of a dangerous ball into the Bournemouth box as Begovic hesitated, with Liverpool continuing to dominate and the home side offering few signs that they could get back into the game.
Fraser fired into the side netting after patient build-up play in which Andrew Surman was prominent, and soon afterwards Bournemouth made another change as Steve Cook came on for Daniels.
Oxlade-Chamberlain was close at the other end, winning the ball, running into the area and hitting the post, and at the other end Mignolet denied Defoe with a diving save.
But with 66 minutes gone Liverpool grabbed their fourth, Coutinho crossing and Firmino diving to glance a header past Begovic.
The outstanding Salah made way for Adam Lallana -- swiftly booked for a rash challenge on Stanislas -- as Jurgen Klopp made a change with 20 minutes remaining, and Coutinho narrowly failed to get a touch to yet another dangerous ball into the area.
Another Liverpool change saw Firmino make way for Dominic Solanke as the game entered its final quarter of an hour, and striker Danny Ings replaced Coutinho with moments left, almost scoring with a low shot that skimmed just wide.