The Toffees went ahead after 24 minutes when Tim Cahill ended a run of 13 months without a goal, but the visitors pushed hard and deservedly equalise through David Goodwillie's bundled effort.
Everton boss Moyes said: "I think we were fortunate to get a point. Blackburn had the better of the first half and we were certainly fortunate to go in 1-0 up but I hoped that once we got it, even though it wasn't one of our better days, we could have hung on to it."
Cahill has been one of Everton's most reliable sources of goals since joining the club in 2004 but he failed to find the net in the whole of 2011. The Australian finally put that right, and Moyes said: "I think he's looked more likely to score in recent weeks. He's been playing better and coming into a little bit of form."
Everton opened the scoring in the 24th minute, albeit in controversial fashion as Marouane Fellaini appeared to handle the ball in setting up Cahill for a tap-in.
Tim Howard missed a lofted ball from Morten Gamst Pedersen in the 72nd minute and Cahill slammed his attempted clearance against Goodwillie to see it rebound into the net.
Both managers had complaints about the goals they conceded, with Blackburn boss Steve Kean feeling referee Mark Halsey's positioning was the reason he did not see Fellaini handle. The Scot said: "If we look back on the game we'll be disappointed because we felt that Everton scored against the run of play.
"I've looked at the goal again and I think it was hard for the ref because he's not in a great position. We saw it on the bench. It came off his chest first and then the ball's been moved off line with his arm."
Moyes conceded Everton got lucky there but felt Halsey missed two offences in the build-up to the equaliser, arguing Steven Nzonzi was offside and that he fouled Howard.
"I've seen a replay and I think our goal was handball but their goal was definitely offside and it's a foul as well," said Moyes.
Source: PA
Source: PA