The Bluebirds had the better of things on home soil but took just one of their chances, leaving them without a win this term and stuck on two points.
A first win against Cardiff in 14 attempts lifts Burnley up to 12th place, while City remain second from bottom ahead of their trip to Tottenham Hotspur next weekend.
After a low-in-quality start to the first top-flight tussle between these two sides since 1962, Cardiff soon found their groove and perhaps should have gone into the interval a goal to the good.
Callum Paterson, restored to the starting lineup as one of four changes for the hosts, dragged a shot wide of goal from a promising position and Murphy then struck the post from a tight angle.
A good stop from Joe Hart down low to his left prevented Kenneth Zohore's close-range prod from crossing the line, and Paterson's headed chance from eight yards was headed off the line from the next attacking wave.
Cardiff also felt that they should have had a penalty when Paterson's shot hit Matt Lowton's hand inside the box, yet the Bluebirds found themselves behind within six minutes of the second-half restart.
Having conceded from the visitors' first attempt of the evening, Cardiff responded well and were on level terms less than 10 minutes later.
Murphy timed his run well to meet the cross from Bruno Ecuele Manga and guide the ball past Hart for his first goal in the Premier League.
The summer signing from Norwich City then had a curled short from 20 yards kept out by Hart as the Welsh side closed in on a quickfire second, only to be dealt a sucker punch at the opposite end.
Some poor defending allowed Gudmundsson to get in behind and loop the ball towards Vokes, who planted a header into the far corner of the net.
Victor Camarasa then shot wide from Cardiff's 17th shot of the contest - compared to Burnley's three at that point - but the Bluebirds could not find a way through for a second time, with Murphy's attempt into the side-netting the closest they came in the remaining minutes.
CARDIFF CITY (4-4-1-1): Etheridge; Bamba, Morrison, Ecuele-Manga, Cunningham; Camarasa, Ralls (Madine 79'), Arter, Murphy; Paterson; Zohore (Ward 71')
BURNLEY (4-4-2): Hart; Lowton, Tarkowski (Long 27'), Mee, Taylor; Lennon, Westwood, Cork, Gudmundsson; Vydra (Wood 72'), Vokes body check tags ::