The Reds returned to the top four with a rampant attacking display, Philippe Coutinho rolling into the near bottom corner after dancing past several Bournemouth defenders.
Dejan Lovren's diving header, after Roberto Firmino's excellent reactions to keep a Coutinho corner in play, doubled their lead before Mohamed Salah scored a sublime third for his 20th goal in all competitions this season.
Firmino scored Liverpool's fourth after the hour, flicking in a header from a Coutinho cross, as the Merseyside outfit extended their unbeaten run to 12 games and leapfrogged Arsenal into fourth.
Midfielder vows there is more still to come from him
Oxlade-Chamberlain told Sky Sports afterwards that "first and foremost" Liverpool "competed right from the start", admitting they knew they "had to do that" and that they "had to be 100 per-cent focused and switched on in defence."
"Then we know when we're going forward that we've got the ability to open doors and score goals," he continued. "The last few games we haven't been able to do that enough but today it came together."
The summer signing from the Gunners, Liverpool's next Premier League opponent, was named Sky's Man of the Match for an industrious and efficient midfield display and insists he is relishing working under Jürgen Klopp.
"I'm trying my best to," he said in response to questions that he is now becoming a 'Klopp player'. "I definitely think that I've made strides forward."
The 24-year-old acknowledged he has "some improving to do" and "areas" of his game to "keep improving on" and said: "There's more to come from me, I know that, but I'm trying every time I get opportunities to do the things the manager wants and also do things that I know I can do. I'm happy with the performance of all the boys today and I thought I did okay today."
He joked that players "usually" have to score a goal to win the Man of the Match award, insisting it is generally "the criteria", before returning to a more serious note and praising his team-mates and saying "everyone did really well today."
"Boys like Phil you know he can come up with magic," Oxlade-Chamberlain said. "Him and and Mo did it again. That's the most important thing. The boys at the back as well, defended well, and that's important for all of us."
"Reds' attacking quartet setting the benchmark"
Liverpool's 'Fab Four' - Coutinho, Salah and Firmino, and unused substitute Sadio Mané - have now contributed 50 goals in all competitions for the season after three of them were on the score-sheet on the South Coast.
Oxlade-Chamberlain hailed: "The four boys up front have been on fire recently. They're not bad at all. It's up to the rest of us when we're all playing, everyone's a quality player, and we've got to try and sneak our noses in there and try and get in and amongst the Fab Four."
He admitted that it is "important" that over the festive period that "when everyone plays you've got to chip in" and that they "can have squad rotation."
"If we can keep doing that and keep getting good results, it keeps everyone fresh and competition for places high," the England international said. "That should push us on as well."