Johannsson, the unpopular former fisheries and agriculture minister, takes over from Sigmundur David Gunnlaugsson and is set to take Iceland into legislative elections expected in the autumn.
He and his cabinet were sworn in by President Olafur Ragnar Grimsson in a ceremony at the presidential residence in Reykjavik.
Johannsson's new right-wing government is under fire from the start, with the opposition set to hold a vote of no confidence tomorrow and pushing for even earlier elections as the country battles its worst political storm since the 2008-9 financial crisis.
Gunnlaugsson quit Tuesday amid mass protests over a hidden offshore account revealed in the "Panama Papers" leak of millions of financial records, becoming the first major political casualty to emerge from the scandal.
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The documents revealed that Gunnlaugsson and his wife owned an offshore company in the British Virgin Islands and had placed millions of dollars of her inheritance there.
Gunnlaugsson sold his 50-per cent share of the company to his wife for a symbolic sum of USD 1 at the end of 2009, but he had neglected to declare the stake as required when he was elected to parliament six months earlier.
Johannsson vowed on the eve of his inauguration that Icelanders would "see the difference" between the two men.