The referee will be the focal point, so we'll start with the controversy. Yes, Rodwell didn't deserve a red. Yes, Liverpool deserved their win regardless.
In fact, Liverpool should have taken an earlier lead, especially since Kuyt is usually automatic for the people from the spot, and the sending off actually helped Everton until the players inevitably tired. Everton are at their best when sitting deep and compact. Moyes does nothing better than organizing his defense, and Liverpool had surprisingly few opportunities despite a man advantage for almost 70 minutes.
Liverpool started the better side, intelligently keeping possession – even if mainly in their own half – to make sure Everton didn't start afire (which is what doomed Liverpool's last away league game). That start resulted in two early chances: Downing on the break centering just ahead of an open Suarez, followed by Kuyt capitalizing on Jagielka's mistake, looping a high cross to Suarez at the back-post but tamely headed straight at Howard.
Everton unsurprisingly responded, winning five corners and two dangerous left-channel free kicks in the first 20 minutes, but Liverpool's set play defense was excellent, with Carra, Carroll, Skrtel, and Lucas dominant in the air against tough opposition. Then came the game-changer.
Martin Atkinson clearly remembered his last Merseyside derby, where he failed to send off Pienaar until the South African racked up multiple dismissal-worthy offenses, and only sent off Kyrgiakos in an incident where both the Greek and Fellaini should have marched. That's the only explanation for Rodwell's straight red. His studs were marginally high, with a trailing leg designed to take a second bite at Suarez's ankles, but it was yellow at best under normal circumstances. Somehow, it's tough to empathize.
As said above, the sending off actually initially helped Everton, as much as a sending off can. Everton knows how to defend with backs against the wall. Liverpool had all the possession, but couldn't carve out opportunities until late in the half. Early balls in Carroll's direction weren't coming off – the strike force failing to link as they had at Wolves – while both Hibbert and Baines sealed off the flanks. Liverpool's first of two chances came on a mistake from Jagielka: a clear, unnecessary penalty taking out Suarez on the left edge of the box. Up stepped Kuyt, so deadly from the spot and in derbies. His low, placed penalty was too easy for Howard: a smart reaction save, but also the Dutchman's worst spot kick. A minute later, Adam pinged an effort off the crossbar from the same location he created an own goal against Wolves, the sixth time Liverpool hit the woodwork since losing to Spurs.
Liverpool kept creating chances after the interval. Carroll had two threatening headers from corners – the first cleared off the line by Saha, the second scrambled behind by Howard – followed by Kuyt placing wide after a quick free kick, but the away side looked increasingly disjointed and frustrated. Finally, substitutions broke that ubiquitous deadlock. With 23 minutes to play, veterans Gerrard and Bellamy replaced derby debutantes Adam and Downing and made all the difference. Gerrard added dynamism, patience, and intelligence in midfield. Bellamy's terrier pace created Carroll's opener.
Everton's increasing fatigue certainly helped, penned back and chasing since Rodwell's exit, but Liverpool still needed an excellent team goal to go ahead. Lucas intelligently spread play to Bellamy on the counter, with both Hibbert and Neville retreating too deep, unable or unwilling to press the ball. Sliding it through to Enrique at the byline, the left-back's cross was cleverly dummied on the bounce by Kuyt (whose run drew both center-backs), warned by Carroll, who blasted on the half-volley for his first league goal of the season. Redemption. Potential realized.
Ten minutes later – with Everton's sole response a quick, tame effort from Cahill immediately after the restart, released by a long flick-on – Liverpool had the crucial second. This one was down to Suarez's innate unwillingness to give up, aided by some baffling defense from the usually reliable Distin. Carra's hoof (and I hesitate to call an accurate long-range pass a hoof, but it is Carra...) found Kuyt between defenders, who knocked down for the Uruguayan. Beating Distin, Baines made a lovely recovery tackle. But with Distin still retreating, the defenders got in a muddle, and Suarez fortunately picked up possession for a point-blank, easy-as-you-like death blow, placed past Howard with his weaker foot. Game, set, match, celebration.
Of course I'd rather focus on Liverpool's resiliency, perseverance, and skill in finally finding the needed goals. It's only the second clean sheet of the season, both surprisingly coming in early kickoffs away from Anfield. Lucas was typically outstanding in a big match (stat line: 68/74 passing – with 40/46 passes forward – 65 passes received, 4/6 on tackles, 2 interceptions, 2/3 on aerial duels); Carragher was untouchable, winning every aerial duel against the dangerous Cahill and Saha; Enrique continued to impress, close to cementing his status as one of the three best left backs in the league; and Kuyt was Kuyt, solidifying the right flank, ensuring Baines rarely threatened with his outstanding crosses, no matter the surprising spot kick mishap. Both Liverpool's strikers scored on their derby debuts, and Dalglish's double substitution made a massive difference. Tactics worked exactly as drawn up, even though it was surprising to see Liverpool stick with the 4-2-2-2 against a packed midfield prior to the man advantage.
But the post-match media narrative will undoubtedly focus on Atkinson's contentious decision, despite subsequent worse fouls by Fellaini, Jagielka, and Hibbert going unpunished. Given past history in derbies, let alone past Liverpool luck with red cards, I'm certainly less than sympathetic. And I'd much rather focus on Liverpool's professional, eventually comprehensive win away from Anfield in one of the most important fixtures of the season.