Kyrgiakos was red-carded for a challenge on Marouane Fellaini after 34 minutes but Liverpool won the game with a Dirk Kuyt header early in the second half.
In-form Everton rarely threatened an equaliser and also ended with 10 men after midfielder Steven Pienaar was sent off for a second yellow card in stoppage time.
Everton were unbeaten in nine league games since losing at home to Liverpool in November but the home side were also on a good run as they seek to claw their way back into the top four.
There was little in the way of good football on show from either side, however, as the game was fiery from the start with tackles flying in from both sides.
"That's what derbies are all about. That's what the fans pay good money for, to come and see passion, commitment and fight," Liverpool captain Steven Gerrard told Sky Sports.
It boiled over after 34 minutes when Fellaini and Kyrgiakos launched studs-up tackles at each other contesting a loose ball.
Like a double-knockout in boxing both men collapsed with shin injuries and both could have been sent off.
However, it was the Greek international defender who saw red while Fellaini was carried off and took no further part.
He was replaced by Mikel Arteta, edging his way back to fitness after a lengthy absence following surgery on a knee injury.
First-half stoppage time produced the two best chances of the half as Steven Gerrard hit the bar with a free-kick for Liverpool and Tim Cahill sent a diving header over the bar from close range.
Liverpool made the breakthrough 10 minutes into the second half when Kuyt headed in a Gerrard corner right under the nose of Everton keeper Tim Howard.
GREAT CROSS
"It was a great cross. When you're with 10 men you won't get many chances, so you have to take everything from the set pieces," Kuyt said.
Liverpool, who had conceded only one goal in their last six league games, then sat deep to protect their lead.
They proved able to do it without too many scares as Everton, despite the promptings of Arteta, rarely threatened in a disappointing display.
The temperature lifted again seven minutes from time when Gerrard caught Pienaar, leading to a mass shoving match and a couple more yellow cards.
Pienaar earned a second yellow and a sending off when his arm caught Gerrard's face, the 10th red card in 11 league games between the two teams.
The win took Liverpool to 44 points and into fourth place for the first time since October.
