OED defines cheeky as “insolent or audacious in address; coolly impudent or presuming” and cites a first appearance in 1859. It was not unknown in the U.S. in subsequent years; a New York Times headline from 1885 read, HOW HILL HELPED PATRICK.; A CHEEKY CIRCULAR SENT THROUGH CHEMUNG COUNTY BY PONY EXPRESS. In 1943, the American writer Max Shulman published a comic novel with the amusing title “Barefoot Boy with Cheek.”
But the Google Ngram below shows that, as with many Britishisms, American only increased significantly in the 1980s.
Presumably, its popularity was helped along by Mike Myers’ late-80s Saturday Night Live character Simon, one of whose catchphrases was “Cheeky monkey!”
Years after retiring, she became friendly with a 37-year-old cheeky chap who had the habit of forging her name on big checks. (New York Times, March 4, 1990). My man Mark Thompson puts up a cheeky post yesterday that I most heartily approved of. (Thomas P.M Barnett, Time.com, August 9, 2011)
notoneoffbritishisms.com 
To me, “cheeky” has a positive connotation, akin to “spunky” in AmE.
Cheeky in American English has a similar connotation to the English use, as in “She’s really cheeky” (she really has a lot of nerve).
We’ll see what influence Scotsman Craig Ferguson now of American late-night TV has, as he regularly calls his audience “cheeky wee monkeys.”
Brits also use “cheeky” to mean “quick” or “brief.” (E.g. “Fancy a cheeky beer on the way?”) I’ve never heard an American use it that way.
Cheeky has come to be a positive thing to say about a child – ‘he’s a cheeky monkey’ now tends to mean, ‘oh isn’t he sweet and lively?’ – and it tends to be said approvingly about lively little boys rather than girls. When I was a kid, if my Mum or my teacher called me cheeky it was really serious – I’d ‘cheeked’ an adult (said something rude or impudent) and there would be hell to pay.
But you can still say ‘the cheek of the man!’ or ‘damned cheek!’ meaning someone has really tried it on with you.You will also hear British workers saying ‘I just popped out for a cheeky fag’, meaning I slipped out for a quick cigarette. They really do say this – I know it might be difficult for US readers to believe.
Mike Myers father is british so it’s not surprising he uses british slang he’s picked up, presumably from his father. “Shag” being one most notable.
I’m English and have enjoyed saying “cheeky” my whole life. “Cheeky bugger!” is a good one. What I’ll never condone is the ridiculous modern usage, such as going for a “cheeky pint”. It sounds contrived and makes me think those who say it are trying too hard to be laddish.