Samba stars help Reds to early lead
Bournemouth couldn't say they hadn't been given a warning sign by Coutinho, who rattled an early free-kick off the post from a tight angle.
He wouldn't miss next time. Slaloming through the Cherries defence after picking up the ball from left-back Andy Robertson, Coutinho checked back slightly at the last to send one final defender the wrong way before placing the ball back across goal and into Asmir Begovic's near corner.
Mohamed Salah was threatening in his usual way, but Liverpool's second goal - six minutes after the first - came from an unusual source.
With a Liverpool corner looking like it was going to be flicked out, Roberto Firmino's strenuous efforts were rewarded as he managed to keep the ball in and divert it back across goal, Dejan Lovren diving in with a good header for his first goal in over a year.
Defoe closest to sparking revival
2-0 up and cruising, Liverpool may have been filled with memories of their last visit to Dean Court, the infamous 4-3 defeat as they blew a 3-1 lead late on.
Bournemouth had a chance to re-open those wounds with their best opportunity of the game coming through Jermain Defoe being sent clean through on goal, but the experienced striker uncharacteristically struck the ball against the frame of the goal and watched it fly back out.
Salah and Firmino seal it
Defoe was punished for that miss just short of half-time as Salah got his customary goal, making it 20 for the season.
Hurtling down the right, Salah cut back onto his favoured left foot and danced around a couple of unlucky defenders before arching the ball into the corner for a third Liverpool goal.
Eddie Howe turned to Ryan Fraser, so influential in last season's win, in search of inspiration.
However, bar a couple of early chances being flashed wide, Bournemouth offered little in the way of attacking threat post half-time.
Liverpool controlled the game better than they have in the past and eventually got a fourth goal, Coutinho's ball in flicked deftly past Begovic via the head of a slightly offside Roberto Firmino.
That effort prompted Jurgen Klopp into getting his stars off, both Salah and Firmino substituted, Liverpool back in form ahead of Friday's trip to the Emirates.