The Reds made most of the running in the game but could not find a goal in a frenetic contest that leaves Rodgers' side five points behind their neighbours with just two games remaining.
"We had our moments," Rodgers told Sky Sports. "We opened the game up and got in a few times. But what was important was that those games can sometimes slip away from you.
"Although Everton caused us few problems in open play, from set-pieces they attack the ball and we dealt with that side of it well.
"That's 13 games we've drawn now. Defensively we've been strong but we just need to find that extra bit of quality.
"We would hope to play better but I can have few complaints as the lads gave everything and the very least we deserved was a point."
The fact that Liverpool did not score and Daniel Sturridge wasted several chances will inevitably lead to suggestions Luis Suarez was badly missed but Rodgers stopped short of admitting as such.
"I suppose people will bring that up because we haven't scored but I thought Daniel Sturridge was outstanding," he added.
"He got in a couple of times and I thought he led the line well. Of course, when you're without Luis Suarez you are without a world-class player."
Everton came closest to finding the breakthrough in the sense that they had a goal disallowed when Sylvain Distin climbed to head home early in the second half.
But when asked about the decision, Rodgers said: "I think as the ball was in the air the referee has already blown for the hands on the shoulders so I thought it was a clear decision."