England fought back from 14-8 down to lead 18-14 but they conceded a last-minute penalty for a needless high tackle on Sonny Bill Williams and Johnson jinked over to level the scores before kicking the conversion to send the defending champions into the final.
They will play arch-rivals Australia next Saturday at Old Trafford, Manchester in a repeat of the last World Cup final in 2008 when the Kiwis pulled off a shock 34-20 win over their neighbours.
"It was a real test match and England had a great game today," New Zealand coach Stephen Kearney told the BBC.
"We kept fighting and hanging in there and with a couple of minutes to go, we took the opportunity. We always back ourselves if there is enough time on the board and I was confident we would give ourselves the opportunity. I am very happy."
England opened the scoring after 15 minutes when powerful surges by forwards James Graham and Sam Burgess set up Sean O'Loughlin to cross the line.
Kevin Sinfield converted the try and kicked a penalty under the posts to put the hosts 8-0 ahead.
New Zealand struck back before halftime, however, thanks to a sublime piece of skill from centre Dean Whare who acrobatically kept the ball in play and flicked it back for Roger Tuivasa-Sheck to plunge over from close range.
Scrumhalf Johnson converted from the touchline and he kicked a penalty just before the interval to send the sides in level at 8-8.
Tuivasa-Sheck claimed his second try four minutes into the second half following a sustained spell of New Zealand pressure and Johnson slotted over another penalty.
England were rocked but they conjured a superb try of their own when a fine flat pass by Sinfield put Kallum Watkins through a hole in the Kiwi defence and the centre dived over to score.
SUSTAINED ASSAULT
Roared on by the Wembley crowd, England sensed their chance and their forwards launched a sustained assault on the New Zealand line before Burgess picked up the ball 20 metres out and stormed over.
Sinfield's conversion put England 18-14 ahead with 11 minutes remaining but, after resisting waves of New Zealand attacks, George Burgess's high tackle on Williams gave New Zealand good field position and Johnson took full advantage.
"It is a tough way to go out in a high level sport which can be cruel at times," England coach Steve McNamara said.
"The two teams went at each other and there was not much in it. We were getting close, doing a lot of things right and a cruel way to finish.
"It has put rugby league on the map and shown people how great it is."
Australia outclassed Fiji, running in six first-half tries by Johnathan Thurston, Darius Boyd, Cooper Cronk, Jarryd Hayne (2) and Josh Papalii to lead 34-0 at the interval.
Centre Hayne completed his hat-trick in the second half and further tries by James Tamou, Boyd, Brett Morris and Andrew Fifita sealed an overwhelming victory for the Kangaroos who have won six of the last seven rugby league World Cups.
(Reporting by Ed Osmond, editing by Josh Reich)
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