The England striker's 85th-minute effort was controversially chalked off courtesy of the linesman's flag, after Blackburn's Brad Friedel had raced out of his goal.
In a game of few chances, a Manuel Fernandes free-kick struck the foot of the post, while David Bentley's set-piece was sensationally saved.
It is now seven Premier League games unbeaten for Mark Hughes' men, but he will have to do without David Dunn and Morten Gamst Pedersen for next week's trip to Arsenal, after the duo both picked up their fifth cautions of the campaign.
Rovers started well and almost opened the scoring with their first foray forward. With less than two minutes on the clock, Bentley swung in a corner from the left and Roque Santa Cruz's glancing header rippled the side of the net.
With ten men in midfield, there was little in the way of attacking action in the first quarter of the game.
Dunn was booked for an attempted challenge on Mikel Arteta, before the Spaniard's solo effort was well saved by Friedel. From the resulting corner, Leon Osman fired just wide.
As a cagey opening period drew to a close, Dunn was fortunate to stay on the pitch for a deliberate handball after some neat skill from Phil Neville.
Bentley then twice fired over from long range, while Phil Jagielka had a header cleared off the line on the stroke of half-time.
Less than 20 seconds after the interval, Bentley's low drive was parried round the post by Tim Howard and, following a quick break to the other end, Arteta's cross fell kindly for the unmarked Fernandes, but Friedel saved well with his legs.
Benni McCarthy was introduced for the home side, but it was Everton who threw bodies further forward.
Osman headed wide from inside the box and, on 62 minutes, Steven Reid was guilty of a foul on Johnson, but Fernandes' free-kick bounced back off the foot of the post.
McCarthy played a majestic one-two with Santa Cruz but, under pressure from Jagielka, the South African striker's appeals for a penalty fell on deaf ears.
Bentley then saw a top corner-bound free-kick brilliantly kept out by Howard, before Moyes' men thought they had won it.
Substitute James Vaughan won the race to the ball ahead of Friedel, his flick inside found Johnson but, after slotting the ball into the empty net, the Englishman was denied by the linesman's flag.
Arteta and Pedersen were both booked for a set-two inside the Rovers half, before the home side had the chance to snatch it.
A quick break saw Bentley centre for Santa Cruz, but the Paraguayan striker glanced his effort inches over the bar, as both sides had to settle for a point - Rovers' third in as many games.