The centre-right opposition accuses the minister of protecting his brother Agron Xhafaj, whom it alleges is still involved in the drug trade.
Eleven police officers were injured when they were hit by stones during the demonstration, Albanian police chief Ardi Veliu told reporters, adding that the injuries had been slight.
A journalist, Blendi Kasmi, also suffered a head injury.
After leaving hospital Kasmi said he had been hit by a police officer using a baton, although other media said he had been hit by an object thrown by demonstrators.
After rallying in front of the government headquarters, where they also called for the resignation of socialist Prime Minister Edi Rama, the protesters moved on to the interior ministry.
There, they threw smoke bombs, as well as eggs and stones, AFP witnessed.
The opposition has accused Agron Xhafaj, who in 2012 received a seven-year jail sentence in absentia in Italy, of continued involvement in drug trafficking.
It recently published a recording in which a man presented as Agron Xhafaj appears to talk to traffickers.
The prosecutor's office said on Thursday it had opened an investigation and had sent the recording to a specialist for verification.
Interior Minister Xhafaj tweeted that if "scientific analysis proves that the accusations of the Democratic party are justified, then I will of course submit my resignation".
The government dismissed the opposition's tactics as "mafia blackmail against the interior minister" who it said was involved in "reform of the justice system with international partners, the US and the European Union".
Speaking at the rally, opposition leader Lulzim Basha accused the prime minister of "collusion with organised crime" and of "plunging the country into misery".
Albanian lawmakers in February passed a law to screen police officers in a bid to clamp down on corruption and links to organised crime, an important part of EU-required judiciary reform.
A candidate nation since 2014, Albania hopes to begin accession talks this year, but the EU has made the fight against corruption and organised crime a requirement for all the countries in the region.
Last year 128 police officers received either administrative sanctions or fines, notably over their involvement in the smuggling of cannabis, a plague in the poor Balkan country.
A former interior minister, Saimir Tahiri, politically allied with Prime Minister Rama, is currently under investigation for his alleged links with cannabis smugglers.
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