Kenny Dalglish's side secure all three points on Scotsman's first trip to Goodison Park since returning to management at Anfield, as defeat piles more misery on David Moyes
The win sees Kenny Dalglish's side move into the top four, whilst defeat for David Moyes' men leaves them in the bottom half of the table, after the away side's second-half brace secured all three points and made up for Dirk Kuyt's missed penalty in the first half.
The early dismissal of Rodwell forced a rethink from Everton and for much of the contest the hosts equalled Liverpool, who couldn't press their advantage until late on in the game.
Both managers made minimal changes to their starting teams, as David Moyes brought Louis Saha back into the fold for his first start of the season, and Kenny Dalglish replaced Jordan Henderson with Kuyt, who has a prolific record in the Merseyside derby.
With the exceptional October sun beating down on Goodison Park, the game began as may have been expected, at full pace with neither side backing down from a challenge.
It was Liverpool who had the first of many early chances to open the scoring, when a delicious ball from Kuyt to Suarez was squandered as the Uruguayan headed tamely into the palms of goalkeeper Tim Howard.
Seconds later an Everton counter resulted in Tim Cahill forcing a fingertip save from Pepe Reina to redirect his superbly timed headed effort over the bar.
As an entertaining encounter unfolded the home side's pressure continued with a series of corners, which resulted in Sylvain Distin narrowly firing over with a deftly curled left-footed shot.
Merseyside derbies are rarely without controversy, and this time it was referee Martin Atkinson's dismissal of Rodwell for a seemingly well timed challenge on Suarez that proved to be the contentious issue.
It seemed a questionable decision at first sight with replays later showing the official may have been too quick to brandish the red card.
Rodwell was furious with the outcome and was seen lashing out at objects on the bench as he made his way to the dressing room, whilst Moyes hurriedly rearranged his plans with his coaches.
The howls of derision from the Everton faithful were aimed firmly at Suarez as Everton looked to calm affairs whilst plan B was implemented.
The Uruguayan soon reinforced his position as public enemy number one when he was fouled by Phil Jagielka late in the first half to give Liverpool a penalty, this time however there was no question about the infringement.
As Dirk Kuyt stepped up to take the spot-kick, Goodison Park held it’s breath - only to release a raucous cheer moments later as the diving low save to the left from Tim Howard kept the scores level.
Liverpool did have one last chance to score before the interval with a long range shot from Charlie Adam smashing into the crossbar.
Handy Andy | Carroll celebrates his first goal of the season at Goodison Park
The deadlock continued with neither team being able to find a way through. Andy Carroll was notably absent in the first half but he was showing a willing to press for the opening goal. Two chances in as many minutes came his way but a Distin lunge and Howard save were enough to keep him at bay.
With Everton still in the game David Moyes chose to reinforce his attack, opting for loanee Royston Drenthe in place of Seamus Coleman as he looked to secure a crucial three points.
Kenny Dalglish followed suit, bringing on Steven Gerrard and Craig Bellamy, with neither manager content with a goalless draw, and Charlie Adam and the ineffective Stewart Downing making way.
Bellamy made a swift impact on the game when his pass to the overlapping Enrique allowed him to break for the byeline and deliver a cross to Carroll who finally broke the resistance of Howard and gave Liverpool a 1-0 lead.
Liverpool sniffed blood in the air and continued to play the ball wide to make the 10 men of Everton chase in the unseasonably warm weather and as Moyes' side tired the visitors took advantage.
A misfired clearance from Distin rebounded off Baines into the path of Suarez, who slid the ball past Howard from six yards out to double his sides lead.
There was no way back now for Everton and Liverpool contently kept hold of the ball and allowed the time to expire.
Moyes would have found it difficult to take many positives from the game but Tim Howard's efforts will not go unnoticed as he was the standout player.