The Swans largely dominated the second half and on-loan Middlesbrough winger Emnes lashed home a low shot from 20 yards to settle the south Wales derby clash.
It was the Dutch player's second goal in four games for the Swans and left misfiring Cardiff, who sorely missed suspended striker Jay Bothroyd, licking their wounds.
Brendan Rodgers' men now lie just three points behind the Bluebirds and showed enough quality to suggest they can mount a sustained push for promotion this season.
Cardiff manager Dave Jones said: "It was a tight game and we made the mistake that has cost us. We will be hung, drawn and quartered now for a week but it's only one game and there is still a long way to go.
"We had good chances but there was hardly anything in it, only half-chances for both clubs and their goal came because our defender was too tight."
Rodgers said: "It was a fantastic result and a wonderful team performance because to come to Cardiff and win such an emotional derby game is superb, both for the players and our fans.
"Cardiff gave us massive respect by changing their style to 4-3-3, so it took us time to settle down.
"When the game is tight, we have the players who can take it on in the second half and our idea of the game was to dominate by playing football."