Mohamed Salah’s 10th goal of the season early in the second period sealed the three points as the European champions continued their seemingly relentless march towards a first title in 30 years.
Defeat for lowly Bournemouth, who failed to produce an attempt on target, was a fifth in succession and leaves them just a single point above the relegation zone.
Meanwhile, Liverpool’s almost-perfect day in Dorset was slightly tempered by a first-half injury to Dejan Lovren.
Reds manager Klopp made seven changes on the south coast as he juggles a packed December fixture list.
The visitors, who had won the previous four meetings between the clubs, scoring 14 goals without reply, dominated first-half possession but had to wait until the 35th minute to take the lead at the end of a simple move which started with Alisson Becker.
After receiving the ball from his goalkeeper via Lovren, midfielder Henderson deceived Bournemouth centre-back Chris Mepham with a long, diagonal pass which allowed Oxlade-Chamberlain to volley his sixth goal of the season beyond goalkeeper Aaron Ramsdale from level with the penalty spot.
The opener had hallmarks of two sloppy goals the hosts conceded at Spurs last weekend and came in the immediate aftermath of Holland international Ake limping off following a superb tackle to deny Salah.
Bournemouth’s players had emerged from the tunnel wearing tracksuit tops with ‘Afobe 9’ on the back in a touching tribute to Benik Afobe following the death of their former team-mate’s two-year-old daughter Amora last week.
The out-of-form hosts have been sucked into trouble at the wrong end of the table after picking up just one win from nine games ahead of kick-off.
The alarming slump looked likely to continue when they fell further behind a minute before the break following a slick Liverpool attack.
Midfielder Keita, making his first league start of the season, collected the ball from Lovren’s replacement Trent Alexander-Arnold and then received it back from Salah’s back-heel before taking a touch and calmly prodding past Ramsdale from 10 yards out.
Without on-loan six-goal top scorer Harry Wilson, who was ineligible against his parent club, the Cherries were failing to test Reds keeper Alisson and unable to haul themselves back into the contest.
Calamitous home defending led to the result being put beyond doubt just nine minutes into the second period.
Inexperienced centre-back Jack Simpson, on in place of Ake, gave the ball away to Keita. The Guinea international then advanced to slip in Salah, who raced away from Mepham to coolly roll home past Ramsdale and atone for a close-range miss in first-half stoppage time.
The three goals came in the space of just 19 minutes either side of half-time.
In the 64th-minute, Wilson became the second Cherries player to limp off, leaving Dominic Solanke, who has now failed to score in 26 appearances since a £19million move from Anfield in January, as a lone striker.
The visitors were content to wind the clock down rather than go full throttle for more goals and convincingly held on for a welcome first clean sheet in 14 games in all competitions.