Martin Atkinson has sent off more players in the Premier League this season than any other referee, and the one he got wrong here, reducing Everton to 10 men, effectively ruined what was developing into a good, competitive Merseyside derby. The red card he showed Jack Rodwell for a legitimate challenge on Luis Suárez left Liverpool with a numerical advantage for three-quarters of the game, yet it still took them another 48 minutes to break down their neighbours' spirited resistance.

Then, watched by Fabio Capello, who names his latest England squad on Sunday night, Andy Carroll scored his first league goal of the season and Suárez, again the best player on the pitch, added the second 10 minutes from the end. The Uruguayan is making good on the investment in him by the club's owner, John W Henry, who had a bad Friday when an accident on his yacht led to a trip to a Boston hospital in a neck brace, but a better Saturday with this win and his being discharged.

The biggest talking point was undoubtedly the dismissal of young Rodwell, midway through the first half, for a ball-winning tackle on Suárez. It was not two-footed, dangerous or malicious, and to the non-partisan eye it did not even look like a foul. It was impossible to gainsay David Moyes, when he insisted: "It wasn't even a free-kick." Red cards are hardly a rarity in these fixtures – this was the 12th in the past 16 derbies – but there can have been none less deserved.

It was, predictably, a game-changing decision. As Kenny Dalglish observed: "Until then it was pretty even at 11 against 11. After they went down to 10, we were very professional."

Everton were always going to be underdogs, even with a full complement. The talk may have been of an Indian summer, but the economy at Goodison is Icelandic in temperature, and while Dalglish has had £114m to spend on reinforcements, Moyes has had to make do and mend, with not a penny available for incoming transfers. Worse still, he has lost his best player, Mikel Arteta, poached by Arsenal. In the circumstances, the paupers raised their game to a degree that reflected great credit on themselves, and in particular their alchemist of a manager.

Initially, both teams showed one significant change, Everton recalling Louis Saha as their main striker, with Phil Neville losing his midfield place in the consequent reshuffle, and Liverpool preferring Dirk Kuyt to Jordan Henderson on their right flank. Unfortunately for Moyes and his charges, their original 4-4-1-1 became 4-4-1 when they were disadvantaged numerically and Saha became too isolated without Tim Cahill backing him up.

Before Rodwell's banishment it had been a good, even game, fiercely competitive as usual, but none the worse for that. Suárez ought to have done better than head weakly at Tim Howard from close range and Cahill, making much better use of his forehead, had Pepe Reina at full stretch in tipping the ball over the bar. Sylvain Distin and Saha also threatened and it was brewing up nicely before the referee's contentious misinterpretation of Rodwell's intent.

Twice just before half-time Liverpool might have taken the lead. First Phil Jagielka brought down Suárez just the wrong side of the 18-yard line and Kuyt stepped up to take the penalty, which he directed low to Howard's left. It was not a bad attempt, but the goalkeeper flew the right way and turned it behind, via the upright. A couple of minutes later Charlie Adam, supplied by Kuyt, shivered the crossbar from 25 yards.

After the interval, Saha cleared a goal-bound header from Carroll off the line and Kuyt shot wide when well placed. Everton, resilience personified, continued to frustrate their more exalted opponents and midway through the second half, with no sign of the breakthrough, Dalglish sent on Steven Gerrard and Craig Bellamy in search of inspiration. Cometh the hour, cometh the Welshman. Within four minutes of his introduction Bellamy charged down the left flank and played in José Enrique, whose cutback from near the byline left Carroll with a routine finish from eight yards. As against Manchester City the previous week, Everton were unable to respond after falling behind and conceded a second when a defensive mix-up between Leighton Baines and Distin let in Suárez for his fourth goal in seven Premier League games.

If it was an unhappy afternoon for Moyes, it was not much better for Capello, less than a week before England's Euro 2012 qualifier against Montenegro. Carroll scored at last, but his overall performance was unimpressive, and Stewart Downing was anonymous. Gerrard got on from the bench, but is not deemed ready for an international comeback and Jagielka blemished an otherwise solid shift by conceding the penalty.

Dalglish said he was "really delighted" by his team, which seemed a little over the top, but such things are relative, and the last time he brought Liverpool to Goodison, in 1991, the harum-scarum 4-4 draw precipitated his resignation.