The win means the Gunners have taken just one point from their first two league games of the season, and are still awaiting their first domestic goal of the new campaign. It brings to an end a miserable week for the club which has also seen Cesc Fabregas finally leave for Barcelona.
Liverpool's first league win away to Arsenal in 11 years put them top of the Premier League table ahead of Saturday's other matches.
A spate of injuries and suspensions meant Arsene Wenger was forced into the uncomfortable position of starting Samir Nasri despite the France midfielder being seemingly on the verge of a move to Manchester City.
In addition to that, defender Carl Jenkinson made his Premier League debut and midfielder Frimpong was handed his first league start.
Matters were made even worse for Arsenal when, after only a quarter of an hour, Laurent Koscielny had to be taken off after suffering back spasms. The Frenchman was replaced by 18-year-old Ignasi Miquel, the only defender named on the Arsenal bench.
After a bright start from the hosts Liverpool began to put Arsenal under more pressure, but both goalkeepers were forced into action and made fine first-half saves.
Frimpong picked up a needless booking after only eight minutes into a niggly first-half display, but he almost drew Arsenal level when he surged forward and drilled a low strike on target which Pepe Reina did well to tip around the foot of the post.
After being subjected to abuse from a section of the travelling Arsenal support during last week's goalless draw at Newcastle, Nasri was treated to a cool reception from the Emirates crowd. If he is close to leaving the club, however, he seemed determined to go out in style. There were several flashes of craft from the playmaker and at one point he picked up a loose ball in his own half and went on a slaloming run through the centre of the pitch. Nasri sold Daniel Agger a dummy and flashed a well-struck effort wide of Reina's far post.
The rain began to hammer down as the second half got underway, and again both sides had decent chances.
Then, with 20 minutes remaining, the Gunners's task was made harder when Frimpong picked up a second unnecessary booking for a tackle on Lucas which was both late and high.
The duo linked up well on 78 minutes to send Suarez charging after the ball into the area. Miquel, who had not looked out of place after being thrown in at the deep end, got his foot to the ball first, but sent it pinging off the chest of Ramsey as the Welshman tracked back and it looped over the helpless Szczesny.
The Liverpool substitutes combined again in the last minute of the allotted 90 when the Portuguese drew Szczesny out of his goal before squaring to Suarez for a simple tap-in.
The coming week for Arsenal could be even worse if things do not go their way. Nasri could well seal his move to City, and they travel to Udinese for the second leg of their Champions League qualification play-off with a slender one-goal lead in the tie before travelling to Old Trafford to face champions Manchester United on Sunday.