Jamal Musiala made the breakthrough in the 22nd minute of the contest, before Ilkay Gundogan doubled the home side's advantage in the 67th minute.
The result has seen Julian Nagelsmann's side, who beat Scotland 5-1 in their competition opener, move onto six points in Group A, enough to secure a spot in the first knockout round with one match to spare, while Hungary sit third, three points behind second-placed Switzerland.
Germany through to round of 16
Fresh from hitting five against Scotland on the opening night of the competition, it was very much a case of same again from head coach Nagelsmann, with the starting XI remaining in place.
Hungary were again headlined by Liverpool star Dominik Szoboszlai, and Marco Rossi's side had a half-chance to take a shock lead inside the first 20 seconds, but Germany goalkeeper Manuel Neuer was out quickly to clear ahead of Roland Sallai.
Peter Gulacsi then had to deny Kai Havertz down the other end, with the Hungary goalkeeper standing strong to prevent the Arsenal attacker from registering in the 12th minute.
Robert Andrich then had one deflected wide from the resulting corner, with both sides having their moments in the final third of the field, but the final piece was missing from the two teams.
That was until the 22nd minute, when Musiala netted his second goal of the tournament, finding the back of the net from inside the box after Gundogan had managed to poke the ball back to the Bayern Munich attacker.
Musiala, Gundogan on the scoresheet for Germany
There was a check for a foul, with Gundogan bundling into the back of Willi Orban, but the referee was happy with the challenge and VAR agreed, which confirmation of the goal sparking more celebrations inside the stadium.
Hungary came close to levelling four minutes later, with Szoboszlai, who was the obvious threat, firing a free kick towards goal, but Neuer was on hand to make the save, and the experienced goalkeeper then managed to deny Barnabas Varga from the rebound.
Szoboszlai then saw one deflected over the crossbar by Jonathan Tah, before Musiala hit one into the side-netting following another excellent passing move; there was also drama late in the half, with Sallai heading into the back of the Germany net, only for the effort to be chalked off for an offside on Orban, who had brought an excellent save from Neuer.
Germany were knocked out in the round of 16 at Euro 2020, while they have failed to make it through the group stage of the last two World Cups, but this team looks capable of challenging for the trophy this summer.
Hungary's goalkeeper did well to keep out a deflected Toni Kroos strike in the 55th minute, before Varga had a brilliant chance down the other end, heading over from close range.
Germany doubled their lead in the 67th minute, with Gundogan sweeping one into the back of the net after being brilliantly found by Maximilian Mittelstadt.
Leroy Sane was denied by Gulacsi late on, and Germany ultimately declared at two, with the tournament hosts, who cleared one off the line in the final exchanges through Joshua Kimmich, becoming the first team at Euro 2024 to advance to the knockout round.
Man of the Match
Kroos was again brilliant in the middle for Germany, with the 34-year-old, who is retiring this summer, showing that he remains one of the best midfielders in world football, while Gundogan also deserves a special mention.
However, Musiala was the star of the show, coming up with the all-important breakthrough, and he almost doubled his tally with a super strike late in the first period.
The Bayern attacker now has two goals at this summer's tournament, and according to WhoScored, he made three tackles and finished with a pass success rate of 87%, putting in a true shift for the team.
The 21-year-old, who was replaced in the 72nd minute, is undoubtedly one of the brightest attackers in world football, and he looks capable of inspiring his team to success at the tournament.
When are Germany, Hungary playing next?
Germany will round off their group-stage campaign against Switzerland on June 23, while Hungary will be hoping to still have a chance of qualification when they tackle Scotland on the same night.