Though Moldova were stubborn in defence, Wales did not lack chances.
They managed to waste every opportunity they had in the opening 45 minutes, however, with Gareth Bale, Ramsey and Robson-Kanu all spurning chances to break the deadlock.
The visitors stepped up their assault in the second half, with Wales manager Chris Coleman sending on 17-year-old Ben Woodburn, who scored the only goal of the game in his international debut against Austria on Saturday, on the hour mark.
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The young Liverpool forward did not get his name on the scoresheet this time but did manage to provide the cross for Robson-Kanu to score the all-important breakthrough goal in the 80th minute.
Moldova twice went close late on, but a strike from Ramsey that cannoned off a defender's foot and arced into the net secured all three points for the visitors.
Serbia top the group on 18 points with Wales, who travel to Georgia next, four points behind and the Irish, who meet Moldova in Dublin before travelling to Cardiff, in third on 13.
(Reporting by Philip O'Connor in Stockholm; Editing by Toby Davis)
