Gibbs said Nancy Killefer, who was on track to become the federal government's chief performance officer, resigned on Tuesday for similar reasons.
Both of them had a series of problems with their personal finances and failed to pay hundreds of thousands in taxes.
Gibbs said both "recognized you can't set an example of responsiblity but accept a different standard of service" during a White House press briefing Tuesday afternoon.
Obama's Treasury Secretary Tim Geithner, however, also had tax problems. He failed to pay $34,000 in taxes while working for the IMF. Gibbs said Geithner's transgressions were not a deal-breaker because "Mr. Geithner has gone through a process..he has gone through the full Senate with bipartisan support."
Gibbs flailed to answer more pressing why the president stood by Geithner despite his tax problems, but not Daschle. He would not answer a question if Obama would have kept supporting Daschle or Killefer if he had not withdrawn his name.
The most he would say is that there are "a few exceptions for people who are uniquely qualified to lead this country."
"I am not going to spend a lot of time up here today looking in the rear view mirror," he said.
Polls: Post-Debate Clinton Can't Get Beyond The Margin Of Error
Watch Live: Trump Speaks in Council Bluffs, Iowa
Watch Live: Trump Campaigns in Waukesha, Wisconsin
The Debate I Heard
200 Reasons Why Hillary Clinton Should Not Be President