Wales, who have 17 points from seven games, can clinch a place at the finals in France if they beat Israel at home on Sunday. Belgium are second on 14 after beating Bosnia 3-1.
Real Madrid forward Bale gave Cyprus keeper Antonis Giorgallides no chance as he powerfully met Aaron Ramsey's cross from the right after 82 minutes to spark wild celebrations between the players, coaching staff and travelling fans.
Wales thought they had taken the lead in the 27th minute when David Edwards headed home a Bale cross but their joy was short-lived as Polish referee Szymon Marciniak disallowed the goal ruling that forward Hal Robson-Kanu had pushed a defender.
Cyprus, harbouring hopes of a third-place finish which brings a playoff spot, matched Wales for most of the first hour before fading but came close to an equaliser after 87 minutes when substitute Pieros Sotiriou's attempt went just wide.
BATTLE HARD
Wales had to battle hard for their first win in Cyprus since 1992 because the home side acquitted themselves well on a hot, humid evening, although were it not for heroics from goalkeeper Giorgallides, Wales winning margin would have been greater.
The first chance fell to Ramsey in the fifth minute following good work from Bale who won the ball in his own half and then embarking on one of his trademark runs, before laying it off to the Arsenal midfielder on the edge of Cyprus' penalty area.
Ramsey then saw his effort agonisingly dip on to the roof of the net.
Five minutes later Giorgallides made a double save, first from Bale's speculative long-range free-kick and then the rebound from Swansea's Neil Taylor who really should have done better from six yards out with the Cyprus goalkeeper on his knees.
Cyprus' only real effort of the half came in the 40th minute when Constantinos Charlamabides' clever free-kick from the right was pulled back to Antreas Makris whose effort went high above Wayne Hennessey's goal.
Cyprus' first shot on target was a long-range effort from Marios Nikolaou in the first minute of the second half, forcing Hennessey into an uneasy save.
Chris Coleman's men almost broke the deadlock in the 49th minute when Bale put Robson-Kanu through but the on-rushing Giorgallides was able to get his hands on the ball before upending the Reading forward.
In the 80th minute Ramsey's shot inside the area was palmed away by Georgallides following good work by Taylor down the left.
Two minutes later Bale's towering header at the far post finally managed to beat the heroic Georgallides.
Cyprus pushed for an equaliser but in the end Wales took all three points to move them ever closer to ending their 57-year wait to reappear in a major finals.
(Eiting by Mike Collett and Ken Ferris)
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