Oxlade-Chamberlain's low strike opened the scoring for the Reds in the 10th minute, before Sane thundered in an equaliser for the league leaders in the 40th minute.
Bernardo Silva and Ilkay Gundogan converted late on, but City had to accept their fifth successive away loss against Liverpool despite becoming the only team has to score a few times in the league at Anfield this season.
After signing for £50m in the summer, Benjamin Mendy lasted just five games before being forced onto the sidelines with a long-term ACL injury. However the defender will be tested against city as Pep Guardiola's men arrives at Anfield to continue their terrific run.
Joe Gomez misjudged Kyle Walker's crossfield pass, which Leroy Sane controlled before cutting in to beat the right-back and Joel Matip before drilling beyond Karius at the near post.
City threatened an awesome comeback after goals by substitute Bernardo Silva, in the 84th, and Ilkay Gundogan, in the first minute of stoppage time, but Liverpool held on in a nervy finale.
The former Chelsea star has netted 18 goals in 23 appearances in the Premier League this season.
The second half was when the game really came alive.
'We will defend our position game by game'.
Clive Brunskill GettyImages
City, who still lead second-placed Manchester United by 15 points, gave almost as good as they got, with Leroy Sane getting a deserved equaliser just before halftime and Bernardo Silva and Ilkay Gundogan ensuring a nail-biting finish.
However, Liverpool vs. City - and more importantly Klopp vs. Guardiola - is the rivalry that every neutral fan and media member should root for to dominate the league.
It was a massive victory for Liverpool which moved them up to third place and sent a massive statement to the rest of the Premier League.
In a tie that ebbed and flowed the home side looked to be more adventurous and paid dividends when the silky Brazilian Roberto Firmino beat John Stones with his tenacity and delightfully lobbed Ederson to restore Liverpool's 2-1 lead.
Sane's second goal in his last 15 appearances should have restored City's equilibrium and they nearly took the lead soon after the interval when Nicolas Otamendi's looping header hit the crossbar.
Six minutes into the second half Nicolas Otamendi's header against the bar was the signal for Liverpool to raise their game.
City will hope to get back to winning ways in the league sooner rather than later as they look to secure their domestic dominance before taking their own attacking football roadshow to the continent in the Champions League.
The match was over as a contest in the 68 minute when a thunderous long range shot from Salah found the back of the net, 4-1. "Today showed they are human and can make mistakes and the others have to try and apply a bit of pressure, but it will be hard".
- UK's May plans Cabinet changes as Brexit enters new phase
- Trump Slams Controversial Spying Program His White House Aggressively Supports
- Delta Air Lines Inc Earnings Advance 13% In Q4
- Man Utd boss Mourinho: Lukaku pushed me to select him for Everton
- Events to celebrate Martin Luther King Jr. Day planned for Metro Detroit
- Elliott DeGeneres, Ellen DeGeneres' father, passes away at 92
- New York City Just Sued 5 Oil Companies. Could Chicago?
- Rajya Sabha adjourned for the day following ruckus over triple talaq bill
- Majority don't want Oprah to run
- President Trump Addresses Record Farm Bureau Crowd