The film, McDonagh's third after the much-admired "In Bruges" (2008) and 2012's "Seven Psychopaths", had critics advising "Fargo" star McDormand to dust off her Oscar acceptance speech after the audience clapped and chortled their way through the film's world premiere.
A surprise-rich screenplay is executed by an ensemble cast that also features Woody Harrelson, Sam Rockwell, Australia's Abbie Cornish and "Game of Thrones" actor Peter Dinklage.
McDonagh, an acclaimed theatre writer who says he prefers making films, wrote the script specifically for the Emmy, Tony and Oscar-winning McDormand, on the basis of an idea that first began to germinate 20 years ago when he was travelling across America by bus.
A decade later, he began to put a back story to a hard-to-explain billboard that had stuck in his mind, based on a mother whose daughter has been raped and murdered.
"Once I had decided it was a mother, the film wrote itself," he said. "And picturing Frances in my mind helped me write it."
- John Wayne attitude -
McDormand, the 1997 Best Actress Oscar winner for her performance in "Fargo", said she had prepared for the role by talking to people who had lost children.
"One of the things I discovered was that if you lose a husband or wife you are a widower, if you lose a parent you are an orphan but if a child dies there is no word for it," she said.
McDormand plays the mother, Mildred Hayes, in a state of semi-unhinged fury over her loss and the absence of any tangible progress in the investigation.
"You're in the ring with a heavy hitter," an admiring Harrelson said of his co-star's performance.
The actress also watched old John Wayne westerns in an attempt to capture the attitude, even the walk, of a character bent on blowing everyone away in a righteous fury, although rather than being a gunslinger Mildred is armed only with a razor-edged wit and a potty mouth.
As a way of channelling her grief-infused rage, Mildred decides to send a hard-hitting message to the local police chief Bill Willoughby (Harrelson), via three disused billboards on a road into town.
Willoughby, an amiable, well-liked figure, is dying of pancreatic cancer.
Mildred is aware but despite pressure from her son (Lucas Hedges), ex-husband Charlie (John Hawkes), now shacked up with an airheaded teenager (Samara Weaving), her erstwhile priest and most of the town, she will not be deterred from her mission to "concentrate their minds some" on her daughter's awful fate.
- Melancholy and funny -
"It wouldn't be as effective after you croak," she tells the sheriff, not long before he dies, unexpectedly soon and in circumstances that the townspeople inevitably link to her billboard campaign.
A backlash starts. Mildred's friends and Red Welby, the freckle-faced and cocky young agent (Caleb Landry Jones) who rents her the billboard space, are targeted by Dixon (Rockwell), Willoughby's intellectually challenged and emotionally stunted deputy, in violent reprisals that cost him his badge.
By this time we know that Mildred is also haunted by what-I-said, and what-I-did-and-didn't-do regrets about the day her daughter left home, never to return.
As the pressure created by her helplessness in the face of her trauma intensifies, Mildred decides on another dramatic gesture to give voice to her grief.
This one goes badly wrong but also indirectly leads to the first breakthrough in the case, giving the sociopathic mummy's boy Dixon an opportunity to redeem himself.
"It is melancholy and funny, that is what Martin does best and that is kind of what humanity is about," McDormand said.
"The script was like a really good piece of literature so that made it a nice ride, really satisfying."
The Venice audience appeared to concur.