Despite playing over an hour against ten man, following Jack Rodwell's controversial red card, Liverpool struggled to carve open a resolute Everton defence, until Andy Carroll broke the deadlock in the 71st minute.
The victory was eventually secured when Luis Suarez added a second for the Reds, and Kelly admits manager Kenny Dalglish told his charges to remain patient during half-time.
"It was brilliant to play in my first derby at their place [Goodison Park]," Kelly said on Liverpool's official website. "It was intimidating at the start with the crowd against us but we stuck together.
"We kept to our game plan which was to get the ball down and play.
"At half-time, Kenny told us not to rush and that if we were patient the chances would come.
"The last thing we wanted to do was lose because we were all up attacking and leave ourselves open to being caught on the counter attack.
"We were patient and in the end the goals came. We deserved the points."
To add to Liverpool's joy, the Reds back four managed to keep only their second clean sheet of the season in all competition.
And Kelly admits defence has been a focus during training in recent weeks.
"We've been working hard in training as a defensive unit.
"We've been trying to get the right balance between attacking and defending so that when we lose the ball we're not too open.
"We did that well against Everton. We're all made up with the clean sheet and now we want to keep more of them."
And fresh from a highly-charged encounter against their neighbours, Kelly is looking forward to another after the international break against perennial rivals Manchester United.
"It's two massive fixtures for us with the derby and then the Man United game.
"As a team, we were all made up with the win [over Everton] and now after the internationals we'll get back together next week and prepare to face United."
Liverpool are currently fifth in the Premier League standings, six points behind joint leaders United and Manchester City.