Ramsey, making his first Wales start in over a year after being dogged by injuries, scored in each half as Ryan Giggs' side leapfrogged Hungary into second spot in Group E.
The Juventus midfielder headed Wales into a 15th-minute lead to settle the nerves of a capacity Cardiff City Stadium crowd.
Ramsey's second just after the restart was his 16th international goal, taking him past the great John Charles on the all-time Wales scoring list and into joint-seventh place.
Victory secured Wales' third appearance at a major tournament, coming just four years after their second when the Dragons reached the semi-finals of Euro 2016.
Wales had the cushion of a home play-off tie, but Giggs' side were unbeaten in five games since losing to Hungary in June and confidence was high that automatic qualification would be achieved.
Hungary had also lost four of their last five matches and arrived in Cardiff weakened by injury and suspension.
Central defenders Willi Orban and Tamas Kadar and left-back Mihaly Korhut were all missing to leave a makeshift Magyars defence worrying what was to come.
Hungary boss Marco Rossi had expressed concern about the quality of Wales' attacking players and Ramsey's welcome return bolstered that department even further.
Hungary started in confident fashion, relaxed in possession and skipper Balasz Dzsudzsak attempting a speculative effort inside 45 seconds.
But Bale had the first on-target effort, straight at goalkeeper Peter Gulacsi, and Wales were ahead within another two minutes,
Connor Roberts slipped the ball down the flank for Bale to cut back onto his left foot and deliver a delicious cross.
Hungary were left counting the cost of those missing central defenders as Ramsey was granted the freedom of the penalty area to nod home his first Wales goal since September 2018.
Botond Barath headed over a Dzsudzsak free-kick but Wales almost doubled their lead from a swift counter attack as Hungary, needing to win themselves to guarantee qualification, showed signs of taking risks.
James, Ben Davies and Ramsey were all involved before Bale sent over a teasing cross which Kieffer Moore put just wide of the post.
Wales appeared comfortable but they received a huge let-off after 33 minutes.
Giant striker Adam Szalai nodded the ball down to present Dominik Szoboszlai with a shooting opportunity from 12 yards.
Wayne Hennessey pushed the effort out and saved Sallai's follow-up to leave boss Rossi wringing his hands on the touchline.
It was to prove a costly miss as Wales doubled their lead within two minutes of the restart.
Moore met Davies' free-kick and his skewed shot fell into the path of Ramsey, who lifted the ball over Gulasci and into the roof of the net from a tight angle.
Bale was inches away from adding another with a free-kick on the edge of the box as Wales piled on the pressure and the Hungary fans reacted by throwing some flares.
Ramsey almost completed his hat-trick after James surged down the left, but Gulacsi stuck out a leg to prevent the ball from arrowing into the bottom corner of his net.
But Wales saw the game out comfortably and the celebrations could begin.