The visitors hit the post through 19-year-old Jordon Ibe, on only his second Barclays Premier League start, but that was the closest either side came.
Everton did not register a shot on target until the 86th minute when Seamus Coleman forced Simon Mignolet to save.
Ibe was named man of the match, which justified Rodgers's decision to last month bring him back from a loan spell at Derby.
"I have been just waiting to put him in
I called him back because I thought now he can really contribute," he said.
"He had a great spell on loan, which is what you want from your young players, and he had a good time at a good club but it was time for him to come back and feature.
"I had no qualms about putting him in
It was a brilliant performance by him, he was unfortunate not to have scored with his run and strike.
"He has wonderful temperament and technique and is tactically very good for a kid coming into that system
I am very pleased for him.
"What was decent tonight was in such an intense atmosphere the composure of the young players.
"Emre Can, he was like a Rolls Royce at the back at 21 and he defended fantastically.
"Jordon Ibe and the rest showed great composure
They connected the game really well and we now have got a real solid structure defensively and 86 minutes was (Everton's) first shot on target."
The downside was the early loss of midfielder Lucas Leiva to a thigh problem, which is likely to keep the Brazilian out for a while.
Everton had their own injury issues to deal with prior to kick-off with left-back Leighton Baines failing a fitness test on a knee injury and winger Aiden McGeady likely to be out for longer after sustaining bruising on a bone in his knee.
"It was a bit of a disappointment
He (Leighton) didn't want to miss the game for any money in the world but the knee wouldn't settle," said Toffees boss Roberto Martinez.
"He wasn't 100 per cent to play
I don't know about Wednesday (against Chelsea)
He could be available
I don't think it is a long-term problem.
"Aiden McGeady got a knock in training and has a bruised bone in his knee and he will be longer."
Martinez felt his side handled a technical game well.
"It became a fascinating game, tactically, two asymmetric shapes that would allow some space," he added.
"There weren't many open chances; Liverpool hit the post in the first half from long range and Steven Gerrard's overhead kick, that was all.
"We were in good positions in the first half but couldn't find the composure or final pass but when Seamus Coleman got into the penalty late in the second half you thought that was the moment.
"I thought the tempo and intensity of the sides was typical of a Merseyside derby.
"We got a very important clean sheet and another strong performance that will allow us to become more confident on the ball and get that arrogance with our play."
The Spaniard defended his handling of Ross Barkley, who was set to come on five minutes before half-time but did not get on until five minutes from the end.
"Steven Naismith had a really nasty fall, his neck got twisted and the doctor was concerned but we felt he could carry on until half-time," he said.
"He didn't want to come off, Mo Besic was carrying a bit of a niggle, James McCarthy was coming back (from injury), Gareth Barry was on nine bookings so I had to make sure I didn't use a substitution cheaply.
"Ross was desperate to come on but Naisy was desperate not to come off and it was the right call medically."
Source : PA
Source: PA