Salah opened the scoring in the first half after reacting quickest to a rebound from a Roberto Firmino shot. However, the Egypt international appeared to be in an offside position.
There was nothing wrong with Salah's two fine solo goals in the second half, which he scored either side of Cook unfortunately turning an Andy Robertson cross into his own net.
Central Role Makes Salah a Greater Threat
Salah made his mark tormenting Premier League defences from wide areas last season. Even so, a central role may be the 26-year-old's future based on how effective he was through the middle against the Cherries.
With fellow wide forward Sadio Mane on the bench, Salah joined Firmino in a two-pronged forward line as part of a 4-4-2 formation. The pair had license to roam, but it was Firmino who most often dropped off the front to leave Salah leading the line centrally.
The Egyptian's pace and deceptive strength occupied both Bournemouth centre-backs. Meanwhile, Salah's clever use of movement also put him into space and made him an easy target for a reshuffled and impressive midfield.
Salah regularly peeled on to the flank to draw a defender out of position before spinning in behind. He also timed his movement well to react to the range and vision of Firmino.
Even so, there was an element of fortune about the way Salah opened the scoring when he slotted beyond Asmir Begovic from an offside position after Firmino's shot had been parried.
It was enough to continue his domination of the Cherries and give Salah a moment of club history:
Salah joined Liverpool's top-10 Premier League scorers with his second goal. His mazy run and classy finish three minutes after the restart effectively ended the game as a contest.
An even better run sent Salah clear on the break 13 minutes from time as he proceeded to tie defenders in knots and baffle Begovic before poking the ball in:
Playing Salah through the middle maintains Liverpool's considerable threat on the break. It also gives those around him, particularly the creative Firmino, an easier and closer target to aim for.
With his ability to traverse every part of the final third from a central spot, Salah is also more tricky for defenders to track. When he's on the flank, the runs from out to in can become predictable.
Keeping Salah central can be Liverpool's ace in the fight for the title.
Keita and Fabinho Partnership Vital to Liverpool's Title Charge
There was a telling moment not long after Salah had opened the scoring that proved the most important partnership in Liverpool's ranks is further back in midfield.
The passage of play involved Naby Keita dropping deep to retrieve possession and get the visitors moving forward. After a successful tackle, Keita sauntered forward on a typical gliding run. As he did, Fabinho dropped off behind him.
This pulley motion was typical of the way Liverpool's two summer imports performed on their first league start together. The obvious fact was Keita's partnership with Fabinho gave the title-chasers the vital balance they have lacked in midfield.
Said balance came from the contrast between Keita's flair and the aggressive resolve and defensive awareness of Fabinho.
Keita showed off his impish ingenuity by the way he carried the ball and stayed bold in possession:
It said everything about Keita's growing importance that Jurgen Klopp was quick to protect him ahead of UEFA Champions League duty against Napoli on Tuesday when he substituted the classy No. 8 after 65 minutes.
Managing Keita's fitness will be key after the way he made those around him, particularly Fabinho, better at Bournemouth.
While Keita wasn't afraid to trust a trick, Fabinho remained content to patrol the base of midfield with destructive intent. The Brazil international broke up play frequently, denying the Cherries their usual threat on the break.
Fabinho's willingness to sit deep and anticipate danger gave Keita and Xherdan Shaqiri the freedom to exploit the pockets of space between the midfield and forward lines.
The blend of silk and steel Keita and Fabinho provided is a common trait in all title-winning teams. Keeping this partnership going offers Liverpool's best means for staying ahead of City from now until the end of the season.
Bournemouth Can't Cope Without Callum Wilson
Liverpool benefited from being able to lean on attacking talisman Salah, but his brace only further highlighted how much Bournemouth missed Callum Wilson.
The England international has been one of the breakout stars of the campaign, but manager Eddie Howe might have feared the worst when he ruled the 26-year-old out:
AFC Bournemouth @afcbournemouth
The boss updates on @CallumWilson's absence... "Callum has a slight hamstring issue, which we don't think is too bad but he misses this week. It's a big blow but Junior comes in, who is a very good player and will give us a lot of quality." #BOULIV // #afcb 🍒 https://t.co/1BS7UufgqU
Howe wasn't the only one left to rue the absence of the prolific striker:
The Bournemouth boss should be concerned after watching his team's tame attempts to cope without the man who has nine goals to his credit in all competitions this season.
Bournemouth missed Wilson's aerial presence. The Cherries also lacked impetus on the counter without the threat of Wilson's pace, as well as his ability to play with his back to goal and link with creative winger Ryan Fraser.
Of all the players hampered by Wilson's absence, Fraser looked the most lost. The Scotland international was given a floating role behind Joshua King but missed the regular target for his astute supply.
While King toiled hard up top, his ability to work the channels and make runs from out to in are more effective when he has a focal point to play off. King's struggles raised further questions about Howe's decision not to give a start to veteran goal-getter Jermain Defoe.
The 36-year-old began on the bench despite his enviable scoring record. However, while Defoe knows where the net is, he's another striker who benefits when partnered with a player capable of holding off defenders.
Bournemouth don't have the strength in depth at the centre-forward position to cope without Wilson. It's a worrying frailty for a team with aspirations of finishing in the European places.
Liverpool will attempt to seal qualification from the UEFA Champions League group stage when they host Napoli at Anfield on Tuesday. Meanwhile, Bournemouth are next in action away to Wolverhampton Wanderers in the Premier League on Saturday.