Uruguay international Suarez was berated as a cheat by the majority of Goodison Park - and appeared to have a coin thrown at him - after making the most of Rodwell's sliding tackle midway through the first half.
He also won a penalty - which Dirk Kuyt missed - to add to his growing reputation as Mr Unpopular with the Blue half of Merseyside.
Suarez, and the performance of referee Martin Atkinson, somewhat overshadowed Andy Carroll's first Barclays Premier League goal of the season.
His strike was the breakthrough and set his side on the way to victory which Everton had worked so hard for so long to deny their visitors.
The last time Kenny Dalglish left Goodison Park as Liverpool manager more than 20 years ago after a 4-4 FA Cup draw the major story developed two days later when he surprisingly quit citing health problems, brought on partly by the aftermath of the Hillsborough disaster.
However, on his first match back as Reds boss, the headlines were being written as early as the 23rd minute.
Rodwell slid in to challenge Suarez and although seemingly winning the ball with a tackle which did not appear to be dangerous Atkinson immediately brandished a red card.
It was the 20th dismissal in 39 Premier League meetings, 13 of which have been Everton players with eight of those coming in the last 10 games at Goodison.
Prior to that Everton had produced the best chances with Tim Cahill - passed fit after struggling with a shin injury all week - forcing Jose Reina into a fingertip save after rising highest at the far post and Sylvain Distin blazing a shot just over.
But momentum of play changed in an instant after the dismissal as Cahill, who had been posing a threat to the Liverpool defence in an advanced role behind Louis Saha on his first start of the season, was dropped back into a left-sided midfield role.
Everton's high-tempo pressing play was abandoned for a deeper-lying, more measured approach and that played into the visitors' hands as they were able to control possession of the ball.
However, they struggled to make a breakthrough until Phil Jagielka brought down Suarez on the corner of the penalty area just before half-time and Atkinson pointed to the spot.
Kuyt, scorer of five derby goals, stepped up but Howard saved brilliantly low to his left - the first miss from the spot in a Merseyside derby since Robbie Fowler in April 2001.
Howard joined fellow Evertonian George Kitchen and Liverpool duo Sam Hardy and Bruce Grobbelaar as goalkeepers who have saved penalties in cross-city meetings.
And while Charlie Adam did beat Howard in added time at the end of the half his 25-yard shot cannoned down off the crossbar.
Everton continued to make light of their reduced number after the break with Saha shooting just wide after being teed up by Cahill just inside the area.
But Liverpool were starting to create more openings and Carroll's header looked to be creeping inside the post from Stewart Downing's corner before Saha hacked away.
Another Carroll effort had Howard scrambling to his left to turn around the post while Kuyt deflected Adam's quickly-taken free-kick wide.
Saha rifled a 25-yard shot just wide of Reina's right-hand post before, in the 67th minute, Dalglish sent on Steven Gerrard, making only his third substitute appearance after six months out after a groin operation, and Craig Bellamy to win him the game.
But it was not the talismanic Reds captain who made the difference but his fellow substitute.
Just four minutes after coming on the Wales international's run down the left released Jose Enrique and Kuyt cleverly ducked under the Spaniard's cross to allow Carroll to power home from eight yards.
The match was wrapped up in the 82nd minute when Suarez grabbed the second.
Collecting Kuyt's knockdown he was lucky to have Distin's clearance bounce into his chest but he made the most of it to place a left-footed shot past Howard from six yards.
The striker's celebrations in front of the Gwladys Street end were interrupted by a number of bottles being thrown onto the pitch.
Kuyt then hit a post before Suarez showed Atkinson a coin which he alleged had been thrown from the stands.
This match has gained a reputation for being the 'friendly derby' over recent years. It was definitely not on this occasion.