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With reference to the act of shaving with a cutthroat razor (which may result in injury), the colloquial expression close shave means: a narrow escape from danger, disaster or mishap.

This phrase was used punningly in the following from The Garstang Courier (Garstang, Lancashire, England) of Friday 14^th January 2000 [page 9, column 3]:

What a cash razor!

One man’s close shave with a hairdresser wielding a cut-throat razor helped raise vital funds for an Oxfam campaign.
The event, arranged as part of the charity’s Give It Up for Ghana Appeal, saw hairdresser Gail O’Brien remove the beard of willing victim, Bruce Crowther—using a traditional cut-throat razor.
The pair, who donned Victorian clothing to coincide with Garstang’s Christmas festival, managed to raise £380.
Miss O’Brien, of Kwik Cutz in Park Hill Road, said: “I learned cut-throat shaving over a period of about six or seven weeks and it took me about 40 minutes in all to shave Bruce.”
Mr Crowther, chairman of the Garstang branch of Oxfam, said: “We raised quite a bit more than we anticipated and the shave was far more pleasant than I’d expected.
“In fact, I think Gail was more nervous than I was.”
Money raised from the event will now go towards funding projects in Ghana.

These are, in chronological order, the earliest occurrences of the expression close shave that I have found:

1-: From “Copy of a Letter from a Gentleman belonging to one of the Discovery Ships, to his friend in Glasgow”, dated “Griper, at Sea, 22d September, 1820”, published in the Glasgow Herald (Glasgow, Lanarkshire, Scotland) of Monday 6^th November 1820 [page 1, column 2]:

Winter set in about the middle of September. A harbour was then most anxiously looked for, which we were fortunate enough to put the ships into by the 26^th September. It was a close shave as to time. The sea, or the lanes of water amongst the ice, which we had hitherto navigated, were now entirely frozen over.

2-: From The Half-Pay Club, published in The U. E. Loyalist (Toronto, Ontario, Canada) of Saturday 12^th August 1826 [page 84, column 3]:

Now stand to your arms, lads—stand to your arms! no nonsense, if you please, the flash from the battery opposite is the signal for sport, and by the powers—ah! that was a close shave! the shot missed you, ’tis true, but it has disabled the tall sergeant that covered you and gone smack through the colours.

3-: From Things in General, published in John Bull (London, England) of Sunday 3^rd June 1827 [page 174, column 3]:

Extract of a Letter from Berbice.—“I returned last night from an excursion up the Corentyn river of nearly 600 miles from Berbice. The Governor (Beard) went to give the annual presents to the tribes of Indians on its banks. I regret to say that he met with a severe accident, the house in which he slept having fallen upon him. It is almost miraculous how he escaped alive, as the whole roof fell down, and some heavy timber struck him on the head.” Mr. Rogers did not hear of this, or he would doubtlessly have called Mr. Beard’s accident a close shave.

4-: From The Nottingham and Newark Mercury (Nottingham, Nottinghamshire, England) of Saturday 3^rd November 1827 [page 3, column 3]:

Who is there that does not know Ben Bower the jolly, hearty, generous old Jehu of Nottingham, who, during his long connexion with the coaching establishments between Sheffield and London, has gained the respect and esteem of all his acquaintance on that line of road—and why should his memory be forgotten? The Judges of Assize are exalted personages—the Sheriff of the County is high in rank; but how often has honest Ben been elevated above them, and taken precedence. The very horses knew him, and acknowledged his restraint. Upon the box he reined triumphant! and many a lift has he given to a fellow creature in distress. He has visited the poles; and talk of a voyage round the world—it is all dickey, compared with the journies he has made. How sweetly he has touched the tits along the road; and at turning a corner, or at a close shave, who could surpass Ben Bower—but his memorial shall be treasured.

5-: From Chester Police Report, published in the Chester Chronicle, and Cheshire and North Wales Advertiser (Chester, Cheshire, England) of Friday 9^th November 1827 [page 3, column 6]:

Mr. Edward Roberts, hair-dresser, appeared to answer to the charge of following his usual occupation on the Lord’s Day.—Roberts did not relish the enforcing of a fine from him, and said, he had no notice to desist from handling his razor and scissors on the Sabbath.—The Mayor hinted that he knew better, and as to notice, he might have seen the bill, in every part of the city, therefore, he must pay the fine, 5s. and 3s. costs.—Tonsor, when in the act of putting the money on the table, shrewdly said, “paying 8s. was rather a close shave.”

6-: From The Trades’ Free Press (London, England) of Saturday 8^th March 1828 [page 260, column 3]:

BRITISH SAILORS IN THE BATTLE OF NAVARIN.
Rev. Mr. Smith’s Address at Freemasons’ Tavern.

The first case related by the Rev. gentleman was that of a sailor, Cruckford [sic], on board his Majesty’s ship Genoa; a shot which killed a man opposite destroyed his eye, and laid the half of his cheek on his shoulder. He lost the use of his senses, and was carried down amongst the dead. By a movement however of his own, towards the wounded cheek, at the moment he was about to be thrown overboard, he was discovered to be still alive. His wound was consequently dressed, but he did not recover the use of his reason, till some time after his arrival at Malta, where it was found that he had not the slightest recollection of the battle, or of any circumstance connected with his sufferings. In the course of his (Mr. Smith’s) conversation with him, Crockford, pointing to the stitches in his cheek, said, “That was a close shave, Sir, wasn’t it ?”