The Dragons created a number of openings but could not find a way through until late on in a match that they simply had to win.
Robson-Kanu converted from a Ben Woodburn cross, before Ramsey sent a deflected shot into the back of the net to seal the three points.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Hennessey; Gunter, Chester, Williams, Davies; Allen, King; Bale, Ramsey, Lawrence; Robson-Kanu
SUBS: Ward, A.Davies, Lockyer, Taylor, Richards, Ampadu, Edwards, Evans, Ledley, Woodburn, Watkins, Vokes
STARTING XI: Cebanu, Posmac, Epureanu, Bordian, Rozgoniuc, Graur, Ionita, Dedov, Anton, Pascenco, Ginsari
SUBS: Pascenco, Namasco, Armas, Carp, Cociuc, Cebotaru, Cojocari, Ivanov, Antoniuc, Platica, Ambros, Bugaiov
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Igor Dobrovolski: "I have a small goal now [in this World Cup campaign] and it is to win. We need to forget the mistakes we have made and look forward. This is a difficult game but... you focus on the next win. I hope the crowd will be the 12th player on the pitch because we need their support. When I was a player and we lost 3-0 people would want to kill me! But these players need support - no matter how they played. I hope a full stadium will be more motivation for them."
Chris Coleman: "[This country] has really improved. It doesn't compare to where we stayed in 1994. We were sleeping in our tracksuits because the beds were so damp. We had cockroaches too. The country was in a lot of trouble at the time. We lost the game and they deserved to win. We let Wales down. That was a tough, tough place. I can see they've come on leaps and bounds. It's very small, tight, and narrow, but a lovely little stadium."
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MOLDOVA SUBS: Pascenco, Namasco, Armas, Carp, Cociuc, Cebotaru, Cojocari, Ivanov, Antoniuc, Platica, Ambros, Bugaiov
WALES SUBS: Ward, A.Davies, Lockyer, Taylor, Richards, Ampadu, Edwards, Evans, Ledley, Woodburn, Watkins, Vokes