Postecoglou named four teenagers in his line-up in Budapest on Thursday and it was an excellent run by 17-year-old Mikey Moore on his full debut which created Pape Sarr's 23rd-minute opener.
It was Sarr's second goal in the competition and after Spurs withstood some late pressure from the Hungarian champions, Johnson wrapped up the points with a left-footed finish with four minutes left.
Ange Postecoglou got warmed up by Spurs' win on a chilly evening in Budapest. (AP PHOTO)
Barnabas Varga set up a tense finish when he volleyed home in the 90th minute, but Tottenham held on to make it five wins in a row after a mediocre start to the season.
Postecoglou made seven changes from the weekend win at Manchester United and handed full debuts to academy graduates Moore and Will Lankshear.
An intimidating atmosphere greeted Spurs' youthful team onto the pitch with the Ferencvaros fans making plenty of noise, but the visitors settled after Varga sent an early effort over.
The influence of Moore, who become the youngest player to play for Tottenham in the Premier League in May, was evident from the start, while Lankshear headed over a Timo Werner cross minutes later.
Ferencvaros had the ball in the net in the 16th minute when Adama Traore headed in Eldar Civic's cross, but it was ruled offside.
Spurs then had to weather a storm with Guglielmo Vicario forced into an outstanding save to deny Varga from close range, although the ball had initially gone out of play.
Moore helped create the opener for Spurs in the 23rd minute after he dribbled past Cristian Ramirez and his attempted ball into Lankshear rolled perfectly for Sarr to rifle in from six yards.
Spurs, creating plenty of chances, should have scored twice in quick succession just beyond the hour mark before Postecoglou turned to senior players, Dejan Kulusevski, James Maddison and Johnson, with 65 minutes played.
Harry Maguire (No.5) scores in stoppage time to earn Manchester United a 3-3 draw with Porto. (AP PHOTO)
In Portugal, super sub Harry Maguire's stoppage-time header saved Manchester United's blushes and potentially Erik ten Hag's job as their Europa League clash with Porto ended in a thrilling 3-3 draw.
Four days after their 3-0 home humiliation at the hands of Tottenham, the Red Devils put criticism of the players and their under-fire manager to the side with an impressive start in Portugal.
Goals from Marcus Rashford and Rasmus Hojlund had United in control, only for Ten Hag's men to lose their way and their defensive faculties as Pepe scored before Samu Omorodion's brilliant brace.
Captain Bruno Fernandes was sent off for the second straight match as Ten Hag squirmed, only for Maguire to head home a Christian Eriksen corner in stoppage time to seal a point.
Lazio also made it two wins out of two by routing Nice 4-1 at home in torrential rain at the Stadio Olimpico in Rome, with Taty Castellano scoring twice for the Italians.
Substitute Tomas Chory headed in a second-half equaliser for Slavia Prague to hold Ajax to a 1-1 draw in Prague, while Anderlecht came from a goal down to win 2-1 at Spanish side Real Sociedad, with Theo Leoni scoring the winner late in the first half.
Ayoub El Kaabi scored two and Santiago Hezze added another for last year's Conference League winners Olympiacos as they beat visiting Braga 3-0, while Germany's Hoffenheim recorded a 2-0 win over Dynamo Kyiv.
Former champions Eintracht Frankfurt claimed a 3-1 win at Besiktas while Rangers went down 4-1 at home to Lyon.
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