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Cardiff scored first half tries from Appleyard, on his first incursion on to the field, and Nick Robinson with the outside-half adding a conversion for a 12-10 half-time lead.
Four minutes in, Argentinian centre Felipe Contepomi put Leinster 3-0 in front from a close-range penalty.
Robinson kicked his first of four misses six minutes later before a loose Blues scrum on 19 minutes saw scrum-half Ryan Powell scurry free and offload for Jim Brownrigg to put Appleyard over for his second of the season.
Full back Donovan van Vuuren found a gap to put Robinson over for his try on 38 minutes under Brian O’Meara’s challenge before Leinster full-back James Norton coasted over in first-half injury time after a good break by centre Christian Warner.
Contepomi’s conversion left two points between the sides and, six minutes after the restart, a sure-footed Leinster attack ended with winger Brendan Burke touching down in the corner for a 15-12 lead.
O’Meara’s penalty, with Contepomi bloodied and off the field, on 62 minutes bolstered the home advantage.
But a luckless Leinster conceded Muller’s effort following a barnstorming break from Canadian back row Dan Baugh.
The visitors got in behind the Lions and, again with Brownrigg in support, Muller got over for Robinson to convert them to their second win in four meetings this season with the Irish province.