Four goals in the first 20 minutes - the first coming after just 52 seconds - left the Gunners reeling. Defender Martin Skrtel scored the first two, with forwards Raheem Sterling and Daniel Sturridge following his lead.
Sterling added his second just after half-time and although Mikel Arteta pulled one back from the penalty spot with 20 minutes to go the match was well over as a contest.
Victory, Liverpool's first in the league over Arsenal since 2007, closed the gap between them in fourth place and the Gunners to five points and they were six off the lead at the end of the day after Chelsea went top.
With Jose Mourinho's side and third-placed Manchester City still to come to Anfield, there is plenty of opportunity to do further damage to the teams above them but Rodgers played down Liverpool's chances.
"You only need to look at our squad really, that can be an indication," said the Reds boss.
"The objective has always been to finish as high as we can but you also have to look at the other teams around you. You only have to look at what Arsenal had on the bench today.
"I want to be in a position where we can really challenge for the title here and the players have been incredible - we just hope to continue with that development and see where it takes us.
"There is no doubt we have been outstanding for a big part of the season.
"We have had one or two off days but in the main our consistency and quality of our game has been incredible.
"We will keep focused and keep our concentration."
Liverpool's display was one of their best for several years and Rodgers was full of praise for what he considered an all-round excellent team effort.
"We were brilliant today, it was a wonderful performance in such a prestigious game between the two teams," Rodgers said.
"The performance level was absolutely out of this world and I am very pleased as it is another marker for us that we are getting better and improving all the time.
"These games are normally very tight games and Liverpool haven't beaten them in the league since 2007 - that is a long time - but I felt we could have gone on and scored more, but I was pleased with how we managed the game.
"The crowd were so excited and they were really wanting us to go for the throat but it was also important we could manage the game as well and that was a pleasing aspect.
"Sometimes with these performances you don't get these types of goals going in but we know we can perform to that level, with Everton (4-0) at home and Tottenham (5-0) away, there are many times we've scored many goals this season."