"I cannot now escape the conclusion that what I have done was in some way wrong even though I did not gain any financial benefit from keeping my relationship secret," Laws said.
In a letter accepting the resignation of a man he appointed less than three weeks ago, David Cameron, the UK's prime minister, described Laws as a "good and honourable man" and said he hoped he could return to the government one day as he had "a huge amount to offer our country".
The Daily Telegraph reported that 44-year-old Laws claimed up to £950 a month for five years to rent a room in two properties owned by Lundie, a lobbyist.
Parliamentary rules have banned politicians from leasing accommodation from spouses, family members or a "partner'' since 2006.
In a statement immediately following the revelations, Laws said he did not consider himself to be in breach of the rules on expenses as he and Lundie had separate bank accounts and separate social lives.
Laws, a virtual unknown outside Westminster before the election, had one of the highest-profile roles in the new government whose priority is to reduce Britain's record 2009-2010 deficit of £156.1bn.
Laws' role made him deputy to George Osborne, the chancellor of the exchequer of the Conservative party, at the Treasury.
He and Osborne this week unveiled spending cuts worth £6.25bn.
Laws successor will be Danny Alexander, another Liberal Democrat, who was formerly the minister responsible for Scotland, Downing Street said.
Cameron has promised to crack down on politicians' abuse of expenses.
Britain's first coalition government since the second World War emerged from the indecisive May 6 general election.