Harry Wilson's stunning free-kick sent Ryan Giggs' men top of their Nations League group following the 1-0 win in Dublin.
Injured Bale returned to Real Madrid after Thursday's 4-1 defeat in Spain and Ramsey was granted time off to be with his wife as she gave birth to their new twins.
The Dragons were also without Ethan Ampadu and Chris Mepham.
It did not matter though, as Wilson's inch-perfect second-half strike proved the difference.
But you have to go back to October 2005 to find the previous occasion on which Wales won without either of their star duo.
And funnily enough it was current boss Giggs who scored twice in that 2-0 win over Azerbaijan.
Wales have played eight games since the start of 2011 in which Bale and Ramsey did not feature.
In that time they managed five defeats and three draws – including a 1-1 friendly draw against Panama last November.
Bale made his Wales debut in January 2016, providing Robert Earnshaw's assist in the 2-1 win over Trinidad and Tobago.
Ramsey made his debut in November 2008, playing 88 minutes of a 1-0 away win in Denmark.
Joe Allen is confident the win in Ireland shows Wales can cope without Bale and Rambo.
He said: "It was important for us to cope without those two players.
"Everyone knows that they're special players, but it's important for us as a team to work to get a result when they're not here."
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