The Reds were caught cold inside the opening two minutes as Antoine Semenyo broke the deadlock, but Andoni Iraola's men went into half time 2-1 down as Luis Diaz and Mohamed Salah turned the game around.
The Cherries would seemingly be handed a lifeline as Alexis Mac Allister was controversially sent for an early bath, but Diogo Jota netted Liverpool's third not long after, as the hosts marched to their first Premier League win of the campaign in chaotic fashion.
Jurgen Klopp included new midfielder Wataru Endo on the bench one day after his arrival, but the Reds boss probably regretted his decision not to fast-track the Japan international into the first XI straight away, as Liverpool were at sixes and sevens during a frenetic start.
Jaidon Anthony capitalised on a disastrous defensive episode to prod home inside two minutes, and while the offside flag came to Liverpool's rescue on that occasion, their blushes would not be spared just one minute later.
The hosts lost the ball in a dangerous area courtesy of a loose touch in midfield by Trent Alexander-Arnold, and while Andy Robertson stuck a foot in before Dominic Solanke could let fly, the ball fell for Semenyo to drill home the rebound into the bottom corner.
A dumbstruck Liverpool came within inches of responding in the sixth minute, but Virgil van Dijk's header from a corner struck the underside of the bar before Bournemouth managed to clear their lines.
An out-of-sorts Alisson Becker was then booked for a foul on Anthony after a dreadful touch presented an opportunity to the Bournemouth attacker - an incident which typified a frenzied opening 10 minutes - but Klopp's team belatedly began to get a grip on proceedings.
After failing with a couple of half-chances, Liverpool finally drew themselves level in the 27th minute, as Alexander-Arnold's sublime pass found Jota, who advanced into the area and tried to cut back for Diaz.
Marcos Senesi got a touch on Jota's pass to slow it down, but the ball still found its way to Diaz, who flicked it up into the air before picking out the bottom corner with an exquisite right-footed volley.
Klopp's side had quickly consigned their poor start to history, and the Reds were awarded a penalty in the 34th minute, as a twisting and turning Dominik Szoboszlai went over Joe Rothwell's trailing leg.
Salah's rocket of a penalty down the middle was kept out by Neto, but the Egypt international needed no second invitation to fire home the rebound, propelling Liverpool into the lead with his ninth goal from nine Premier League games against Bournemouth.
While there were no early goals in the second period, shots were aplenty, as Semenyo evaded several challenges before firing over in the 50th minute, while Jota and Salah squandered good chances of their own not long after.
The on-field pandemonium continued in the 58th minute, as Mac Allister caught Ryan Christie on the ankle with his studs as both men challenged for the ball, and referee Thomas Bramall reached straight for the red card.
However, the Argentine's controversial sending-off - the first of his Premier League career - only seemed to galvanise the 10 men of Liverpool, who extended their lead in the 60th minute as Jota prodded home from close range after Neto spilled a Szoboszlai drive.
Alisson did not cover himself in glory inside the opening 10 minutes, but the Brazilian was Liverpool's hero in the closing stages, making two crucial stops from Hamed Traore in injury time as the Cherries were put to the sword.
Liverpool have a week to recuperate before an intriguing trip to Newcastle United next Sunday, while Bournemouth host Tottenham Hotspur in next Saturday's lunchtime kickoff.