Lacklustre defending allowed Sarr to send Tottenham ahead in the 23rd minute of the contest, with the midfielder sweeping into the bottom corner after Ferencvaros failed to deal with a loose ball inside the box.
Tottenham dominated the possession in the opening half of action, having 62% of the ball, while they had seven attempts, with three of those on target, but Ferencvaros were carrying a threat of their own.
Ferencvaros were pushing forward, as expected, in the second half in a bid to secure a share of the spoils, but the hosts lacked a cutting edge in the final third of the field throughout the contest.
Johnson scored for the fifth game in a row in the 86th minute, but Tottenham had to survive a late scare in the European clash, as Barnabas Varga netted in the 90th minute to give the home fans something to cheer.
The victory is another huge step in the right direction for Tottenham, with the North London outfit now triumphing in their last five matches in all competitions, including a 3-0 success over Manchester United in the Premier League last time out.
The success has also moved Spurs into second position in the Europa League table, boasting six points from their opening two games, while Ferencvaros are down in 32nd, having lost their opening two games of the competition.
FERENCVAROS VS. TOTTENHAM HIGHLIGHTS
Pape Sarr goal vs. Ferencvaros (23rd min, Ferencvaros 0-1 Tottenham)
Tottenham make the breakthrough in the 23rd minute of the European affair, and it is Sarr on the scoresheet.
Following some poor defending from the home side, the ball falls kindly for Sarr inside the penalty box, and the midfielder makes no mistake from close range to break the deadlock.
Brennan Johnson goal vs. Ferencvaros (86th min, Ferencvaros 0-2 Tottenham)
Tottenham double their lead in the 86th minute of the match as Johnson fires into the bottom corner to surely secure all three points for the visitors, who have that winning feeling at the moment.
Barnabas Varga goal vs. Tottenham (90th min, Ferencvaros 1-2 Tottenham)
Late drama! Ferencvaros get their name on the scoresheet in the final minute of normal time, with Varga turning a dangerous cross from Cristian Ramirez into the back of the net from close range.
MAN OF THE MATCH - PEDRO PORRO
Sarr deserves a mention in this category, but the man-of-the-match award does to Pedro Porro due to his overall performance for Tottenham on Thursday night.
The 25-year-old came close to scoring, hitting the woodwork, while he had five successful dribbles and managed 60 accurate passes, with Porro a key contributor to his side's success in Hungary.
FERENCVAROS VS. TOTTENHAM MATCH STATS
Possession: Ferencvaros 43%-57% Tottenham
Shots: Ferencvaros 12-17 Tottenham
Shots on target: Ferencvaros 5-7 Tottenham
Corners: Ferencvaros 4-11 Tottenham
Fouls: Ferencvaros 10-13 Tottenham
BEST STATS
WHAT NEXT?
Tottenham will now switch their attention back to the Premier League, with Ange Postecoglou's side preparing to head to Brighton & Hove Albion on Sunday afternoon.
Ferencvaros, meanwhile, will be aiming to return to winning ways when they take on Paks in Hungary's top flight on Sunday evening.