Although used in an unusual way, and in an unusual structure, I wonder if this comes from the common idiom:

"take the edge off
Ease or assuage, make less severe, as in That snack took the edge off our hunger , or Her kind manner took the edge off her refusal . This term alludes to blunting the edge of a cutting instrument. Shakespeare used it figuratively in The Tempest (4:1): "To take away the edge of that day's celebration." The precise wording of the idiom dates from the first half of the 1900s."

The idea being transmitted is that any formal uneasiness felt about the room, was taken away by the pleasant aroma of garlic. Maybe the smell made it more homely and easy to relax in.

There is a literary meaning of "edged off" that means pushed out of the way or more literally kept out, stopped at the edged or something akin to that.

However in this use I think the author is using both a very literal meaning, as if in a painting or tapestry, to mean with a smell of garlic at the edges - as if there were a visual image of the garlic, and the more normal literary meaning would also be heard.

The more normal literary meaning, I think, is that the air of fancy, expensive diamonds is "edged off" but the thick peasant smell of garlic. I am suggesting "peasant smell" because of the author (O'Henry's) use of American pejoratives for South American leaders/banana republics.

Here are a few examples of the more normal literary use of "edged off":
HEADLINE FROM THE WASHINTON POST
The Rant: Proven plays, major writers often edged off D.C.’s marquee

By Nelson Pressley, Published: July 13The Washington Post
It is not the job of Washington theater to be New York lite. There is no reason for our top-flight professional stages to simply replicate last year’s Manhattan hits."
http://www.washingtonpost.com/entertainment/theater_dance/th...

FROM A NEWS blog
Malaysian crude palm oil futures edged off a one-month high on Thursday, as traders turned cautious over demand prospects, with the worst drought in the U.S. Midwest in 56 years pushing oilseed prices higher. Palm oil futures have been riding on the back of weather-fuelled gains in the soybean oil market after a long weekend holiday and strong export demand, but traders said the price rally might not be sustainable.
http://www.plantationassetmanagement.com/malaysian-crude-pal...
“The market looks a little toppish. The current high prices will certainly hurt demand,” said a trader with a domestic commodities brokerage in Malaysia.

I think that is what is meant here, with neither approval or disapproval of garlic in this phrase.

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Note added at 2 hrs (2012-10-15 14:18:30 GMT)
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Ooops: neither, nor!

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Note added at 2 hrs (2012-10-15 14:26:04 GMT)
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In support of this understanding of "edged off with":

paperspast.natlib.govt.nz/cgi.../paperspast?a=d...
"It ascends in" conical form, and is edged off with fretwork of curious design, and ... "

news.google.com/newspapers?nid=1338&dat...id...sjid...
"... edged off with shirred velvet to match trimmed with bead ornaments; ... "

chroniclingamerica.loc.gov › ... › September 25, 1878
"When Dandy Ferguson first came to this camp he was a gilt-edged ..... The latest carriage parasols are made of Cardinal satin and edged off with deep white ..."

Note the date on the above reference.

www.ukauctioneers.com/lots/963638/lot_enquiries/new
"... the words Souvenir of the Great War edged off with lace ..."





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Note added at 2 hrs (2012-10-15 14:36:14 GMT)
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Of course, "edged off" can also mean slowly retreated from. However, that is not the meaning when it takes the preposition "with", except in e.g. He edged off with his head bowed. The idea of edging off from something is always understood when it has that meaning, even if the preposition "from" is not actually used.

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Note added at 2 hrs (2012-10-15 14:50:10 GMT)
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As gallagy notes, "a finishing touch" is usually positive. However, here are a couple of examples of it being used negatively. It can also be used ironically.


'I confess that I was calm as I saw my last hour approaching, but the thought of broken legs and hands appalls me. To be a cripple would put the finishing touch to me.”'
http://www.chopinproject.com/chopin-quotepage/quotes-by-chop...

"Oh yes, and those dreadful shouting vocals as finishing touch. No, thanks."
www.thoughtsofmetal.com/thunderkraft-totentanz/

I think the garlic smell is considered negative here so it is on top of the other smells or an additional unwanted smell (most people hate the smell) so the

atmosphere is "international" with the smell of the Cuban cigars,
the South American brunettes with presumably their aroma of perfume (and the diamond rings so bifg they're lighting up the room) and over all of that is the faint smell of garlic (which would be typical smell of Latin cooking but not completely masked here by the smoke and perfume

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Note added at 2 hrs (2012-10-15 14:34:38 GMT)
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it's actually making fun of the assembled company and venue in a way: they're all dressed up and acting like rich people but still there's this whiff of garlic hanging off them that all the expensive colognes and perfumes have not been able to mask/disguise/remove completely

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Note added at 3 hrs (2012-10-15 14:56:10 GMT)
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I think, in fact, that this is quite tongue in cheek or, as BDF notes, ironic