The departure of the Latvia's longest-serving premier brought down its centre-right government, a measure of the scale of the political uproar triggered by the tragedy in Riga.
President Andris Berzins said in a statement he now planned to appoint a new government this year. Political analysts said that would stave off the chance of a snap vote earlier than the national elections that were already scheduled for next year.
Dombrovskis did not give details on his reasons for quitting, but his economy minister, Daniels Pavluts, this week partly blamed the collapse on a lack of government oversight of construction projects.
Latvia's government also abolished a national building inspectorate as part of austerity measures that helped pave Latvia's way into the single currency.
"I announce I am resigning from the post of prime minister, taking political responsibility for ... the tragedy," Dombrovskis told journalists, according to local news agency BNS.
The prime minister's spokesman Martins Panke confirmed the resignation to Reuters.
The cause of the collapse remains unknown, though police have opened a criminal investigation focusing on the construction of the building.
Local media said workers had been building a roof garden on the supermarket, a single-storey building about a 30-minute drive from the centre of the capital.
The prime minister had said the supermarket disaster shattered Latvia, a former Soviet republic which joined the European Union nearly a decade ago. President Andris Berzins called the disaster "murder".
Dombrovskis had been in office since 2009 and led the country through sweeping spending cuts and out of the its worst economic downturn since independence.
The country was widely praised for its recovery and is due to join the euro zone on 1 January.
Analysts said they did not expect Dombrovskis' exit would disrupt Latvia's entry into the euro, though polls suggest most voters oppose joining the single currency.
"I don't think there is any impact on the euro zone (entry). The decisions have been made," said Martins Kazaks, Swedbank chief economist in Latvia, noting that the current coalition might also return to office.
Latvia will become the 18th member of the euro zone. The Baltic state has the highest economic growth rate in the EU after a deep recession during the 2008 financial crisis, when it kept its currency pegged to the euro.
Latvia pursued a policy of spending cuts coupled with redundancies and wage cuts that wiped out a fifth of its GDP. Its public debt is now around 41% of GDP, far below the European Union average.