In a game which witnessed crowd trouble and 12 yellow cards brandished, Raul Rusescu gave the hosts a deserved lead on the stroke of halftime, putting the Hungarians, who were beaten 2-1 at home by Northern Ireland in their group opener, under even more pressure.
Yet the visitors breathed a sigh of relief when Dzsudzsak gave keeper Ciprian Tatarusanu no chance with his precise shot eight minutes from time.
Romania, who began their campaign with a 1-0 win at Greece, are top with four points from two matches but Finland and Greece, who have three points each, play their matches later on Saturday.
There was an electric feel at the National Arena, with both teams' vociferous fans creating a deafening roar and letting off dozens of firecrackers and flares shortly before kick off.
Following their shock opening defeat, Hungary's players had decided to make it up to their fans by buying them 1,200 tickets to the match in Bucharest.
Tempers boiled over in the stadium throughout the eagerly-anticipated encounter, with fans warned by the stadium announcer about their behaviour on several occasions and riot police spraying tear gas to dampen crowd trouble.
Scottish referee William Collum also had a busy night policing affairs on the pitch, needing to show 12 yellow cards.
The home team's 4-2-3-1 formation, featuring Rusescu up front, took the early initiative and enjoyed more possession with Lucian Sanmartean working hard to pull the strings in midfield.
However, Romania coach Victor Piturca, already without injured striker Ciprian Marica, was forced to make an early substitution with centre back Dorin Goian replaced by Florin Gardos after four minutes.
Hungary's Adam Szalai had a goal disallowed for offside six minutes before the break, causing visiting fans to set several seats on fire.
Rusescu put the home side in front with his first international goal with an easy finish after some good work by captain Razvan Rat, playing his 99th match for Romania, on the left flank.
Rusescu also had a penalty shout waved away and a goal disallowed for offside in the second half though Hungary almost stole the win soon after Dzsudzsak's strike when substitute Nemanja Nikolic sent a shot just wide.
