Obama, who arrived in Washington on Sunday evening in preparation for his January 20 inauguration, said he had accepted Richardson's withdrawal from the post with "deep regret".
"It is a measure of his willingness to put the nation first that he has removed himself as a candidate for the cabinet in order to avoid any delay in filling this important economic post at this critical time," he said.
The US president-elect faces a busy few days before he formally takes office, including meetings with congressional leaders to try to promote an economic plan that his top spokesman said was "very, very unlikely" to be ready by his January 20 inauguration.
With the US economy hit hard by the global downturn and unemployment skyrocketing, many Americans are looking to the incoming president to help restore confidence and create jobs.
Obama has also drawn criticism for not commenting on Israel's offensive in Gaza.
However, aides have said that while Obama is not officially president he must leave US diplomacy to the incumbent, George Bush.
The Democrat-controlled US senate is due to begin vetting Obama's cabinet nominations this week and observers said an extensive public discussion of the Richardson case could have embarrassed the Obama camp.
The incoming administration has already had to distance itself from a scandal involving the Democratic governor of Illinois who is facing investigation on charges he tried to sell Obama's former senate seat.
Richardson, who was also an energy secretary under Bill Clinton, the former US president, is in his second term as New Mexico's governor and is one of the country's most prominent Hispanic politicians.
The 61-year-old has also led a number of delicate diplomatic missions and was believed to have been on the shortlist to become the next secretary of state.
That post was ultimately offered to Hillary Clinton, Obama's former rival for the Democratic presidential nomination.