England fought back from 14-8 down to lead 18-14 in the final minute but they conceded a 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.
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.
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.
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.
Australia play Fiji in the second semi-final later on Saturday.
(Reporting by Ed Osmond, editing by Josh Reich)
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