Steve Cooper’s side trailed to Demarai Gray’s early penalty at the City Ground on Sunday before Johnson popped up with the equaliser.
Abdoulaye Doucoure put Everton back in front, but Johnson rescued Forest when he netted again late in the second half.
Forest are unbeaten in their last nine league home games and sit four points clear of the relegation zone.
“Both teams were going for it but the referee has had too much involvement really,” Cooper said.
“It’s a pity things like that are affecting games. Everton have done alright out of the referee today.”
Everton sit third bottom, below Leeds on goal difference, as they fight to avoid playing in the second tier for the first time since 1954.
Everton boss Sean Dyche has won just once in five games since starting his reign with a shock victory against Premier League leaders Arsenal.
Dyche’s side were thrashed 4-0 by Arsenal on Wednesday in a rematch that laid bare the brutal truth about his lowly club’s perilous position.
“There are very pleasing signs to what the players are delivering and how they are taking these games on,” Dyche said.
“We need points of course, but there has to be a plan, a method and reason of getting them.”
– Johnson strikes –
Everton kicked off as the lowest scorers in this season’s Premier League and once again, they were without injured England striker Dominic Calvert-Lewin.
Despite Calvert-Lewin’s absence, Everton took the lead after 10 minutes when Jonjo Shelvey tripped Dwight McNeil, conceding a penalty that Gray converted with ease for his first goal since January 3.
It was only Everton’s seventh away league goal this season but they couldn’t hold the advantage for more than nine minutes.
Morgan Gibbs-White’s low strike was parried away by Jordan Pickford, but Johnson was perfectly placed to slot home from six yards.
Everton responded impressively to that blow and captain Seamus Coleman appealed in vain for a penalty after Jack Colback’s challenge sent the defender sprawling.
Dyche’s men regained the lead in the 29th minute when Doucoure headed past Forest keeper Keylor Navas from close-range after Everton won a series of aerial duels from Pickford’s free-kick.
Three of Everton’s four goals under Dyche have come via a set-piece.
They nearly notched one without the aid of Dyche’s set-piece planning just before half-time when McNeil’s dipping volley was tipped over by Navas.
As the tension mounted in the second half, both teams resorted to some ferocious fouls that left referee John Brooks with no option but to issue a string of yellow cards.
It was Forest who emerged with their composure intact and they salvaged a point in the 77th minute.
Ryan Yates’ pass found Johnson on the edge of the area and the Wales forward advanced unhindered by the sluggish Everton defence before curling a fine finish into the top corner from 12 yards.
Forest pushed hard for a winner and Pickford had to save well from Andre Ayew’s low drive before Johnson fired just wide.
AFP
