The killing is thought to be the first of its kind in Europe's worst refugee crisis since World War II.
Tens of thousands of people fleeing war and grinding poverty have been trying to reach Europe by land and sea.
The fatality was among a group of 54 migrants spotted by a patrol near the southeastern town of Sredets close to the Turkish border, Bulgarian interior ministry official Georgy Kostov told a press briefing Friday that took place near the scene of the incident.
Read: Bulgaria charges three men over dead migrants in lorry in Austria
The man died on his way to hospital after being wounded by a bullet that ricocheted when guards fired warning shots, Kostov said.
The migrants "did not obey" a police order to stop, the official said.
"None of the migrants were armed, but they put up resistance."
Patrol officers had fired in the air and "a migrant was injured by a ricochet - according to the testimony of one of the three police officers - and succumbed to his injuries on the way to the hospital," he said.
The migrants said they were Afghans but had no papers, Kostov added.
The spokesman for the UN refugee agency (UNHCR) in Bulgaria, Boris Cheshirkov, called the incident "very regrettable".
"This plan for barriers, fences and police cannot solve the problem of desperate people," he told AFP, recalling a UNHCR appeal to Bulgaria, launched last spring, not to return migrants.
He said it was the first case of a fatal police shooting of a migrant on the EU's borders.
The incident prompted Bulgarian Prime Minister Boyko Borisov to fly home from a summit of European leaders in Brussels where the agreement with Turkey was announced.
Under the plan, Turkey agreed to tackle people-smugglers and take measures to keep more of the millions of refugees fleeing the Syrian conflict from crossing by sea to Europe.
In exchange, European leaders agreed to give Ankara more funds to tackle the problem and ease visa restrictions on Turkish citizens travelling to Europe.
The deal was reached after European Commission officials visited Turkey on Thursday in a last-ditch effort to persuade the government to back the plan, following a visit by President Recep Tayyip Erdogan to Brussels last week.
European Council President Donald Tusk hailed the pact as a "major step forward" but stressed "an agreement with Turkey only makes sense if it effectively contains the flow of refugees".
Read: One-year old boy dies on migrant boat from Turkey to Greece
In response to the Turkish border shooting, Tusk said: "It shows how important our discussion was. Prime Minister Borisov is aware that we are ready to help."
Turkey is the main departure point for the more than 600,000 migrants who have entered Europe this year, most of them making the short but dangerous sea crossing to the Greek islands, but some also coming by land.
The crisis has already claimed the lives of over 3,000 migrants this year who drowned in the Mediterranean trying to reach the continent in makeshift boats.
The 28-nation European Union has been left more divided than ever by the mass migration, especially given fears the passport-free Schengen zone could collapse as countries try to curb the huge numbers of migrants.
A member of the EU but not of Schengen, Bulgaria is on the fringes of the main flow of migrants heading to western Europe through Greece, Macedonia and Serbia.
The country has however seen tens of thousands of migrants transiting since the beginning of the year.
In a move to buttress its porous 260-kilometre (160-mile) border with Turkey, Bulgaria built a 30-kilometre razor wire fence along part of it and dispatched some 2,000 border guards, police and army to guard the rest.
Hungarian Prime Minister Viktor Orban, who has already faced criticism for his hardline stance towards migrants, announced Thursday his country had completed construction of a fence along its southern border with Croatia to stem the massive daily influx.
Croatia said more than 4,800 people had entered on Wednesday, bringing the overall number of arrivals in the EU member state to nearly 175,000.
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