Three goals in the space of nine minutes just before half-time put the hosts clear in front at the Emirates Stadium, the Reds having Emre Can sent off after a consolation panelty
In a match in which the Reds often looked too cautious and unable to take control, Gunners right-back Hector Bellerin took advantage of slack defending from Alberto Moreno to put the hosts in front.
A free kick from Mesut Ozil compounded the Reds' troubles just three minutes later and Alexis Sanchez's powerful finish to a slick team move moments before the break left Liverpool crippled.
The visitors got one back in the second half when Jordan Henderson scored from the penalty spot, Raheem Sterling having been brought down by Bellerin, but they could not push on to save the match.
All that Liverpool got for their troubles in the final stages was a second yellow card for Emre Can, leaving them to close out their defeat with 10 men, and their numerical disadvantage was punished when Olivier Giroud continued his fine recent form by striking a fantastic fourth at the death.
And yet the script could have read so differently if Lazar Markovic had found his passing range in the 20th minute, when he failed to supply the ammunition for Sterling - starting up front in place of Daniel Sturridge - to net what would have been a simple finish.
Sterling dominated the headlines in the build-up to the match, having publicly discussed his contract situation at Liverpool and declared himself "flattered" with links to Arsenal.
But it was the Gunners who started much the brighter as the visitors nervily conceded possession in dangerous areas early on. Joe Allen was the first to err inside 30 seconds, giving the ball away when facing his own goal, with Alexis ultimately shooting wide from 20 yards.
Simon Mignolet was called upon to save from Santi Cazorla and then Aaron Ramsey, the latter having been sent clear by former Arsenal man Kolo Toure's mistake.
Liverpool's moment came and went when Markovic drew the attention of David Ospina, only to fluff his lines in playing a simple square pass for Sterling - the ball narrowly eluding the forward with only the unguarded net in front of him.
It heralded a promising spell of play for Rodgers's men, in which Sterling fired wide and Henderson was agonisingly close to collecting a cross from the 20-year-old inside the box.
Having failed to make that pressure count, Liverpool were punished three times in quick succession. First, Bellerin cut in from the right to curl a shot past Mignolet, who then saw Ozil's precise free kick sail into the same corner three minutes later.
And the match as a contest was effectively ended when Alexis rifled home from the edge of the box, with questions to answer for Mignolet, who appeared to be caught flat-footed.
Sturridge was introduced for Markovic at the break, but Arsenal continued to force the issue - the in-form Giroud's header bringing an excellent save from Mignolet.
Ospina denied Can at the other end before Sterling was hacked down in the box by Bellerin, who was fortunate not to be shown a second yellow.
Henderson just squeezed his 76th-minute spot kick past Ospina but Liverpool's hopes of a comeback were dashed when Can lunged in on Danny Welbeck to earn his marching orders, with Giroud's 10th goal in as many matches wrapping up an excellent afternoon for Arsenal.