It is hoped the deal will end the deadly political rivalry that has left more than 1,000 people dead since presidential elections in December.
Violence erupted after both main parties claimed victory.
The new laws are a key part of enacting a power-sharing deal between Kibaki, who leads the Party of National Unity (PNU) and Raila Odinga, leader of the rival Orange Democratic Movement (ODM).
The Constitution of Kenya (Amendment) Bill was first to be passed on Tuesday, by a unanimous 200-0 majority.
The amendment created the positions of a prime minister and two deputies in the cabinet.
The newly-created prime ministerial post has been designated for Odinga.
The next step was to pass a new law creating these posts in a new unity government, and set out the terms for power sharing in the cabinet.
The law says that the governing coalition between the PNU and the ODM can be dissolved should should one of the partners withdraw.
Odinga's party and Kibaki's coalition will each name a deputy prime minister, while the cabinet will be split evenly between both sides to form a unity government.
However no indication has been given as to when that will happen.
As well as restoring stability and getting the economy back on track, the new administration must also come up with plans for a long term constitutional solution to Kenya's problems.
The violence laid bare underlying political issues of power, inequality, tribal feuds and land ownership.
The crisis began after Odinga accused Kibaki of stealing the election in December, but the violence also opened up tribal rifts going back many decades.
As well as more than 1,000 people dead, the fighting left at least 300,000 more homeless.
Kibaki said earlier on Tuesday that the two bills held the chance to resolve the political crisis.
"We are now confident a permanent solution to the crisis will be achieved," he said.
Odinga said MPs should continue to work towards reconciliation and support a drive to address key underlying issues.
"A time has come for us to hold a national ethnic conference where we will have representatives from all the 42 tribes in the country come together to discuss openly how we want to lead this country," he said.
Kibaki called for a full constitutional review for Kenya, more than two years since a government-backed draft was rejected by a referendum.
"Putting in place a new constitution would form the basis for building a new Kenya and facilitate the country to join the new world," he said.
The agreement between Kibaki and Odinga was reached after weeks of mediation by Kofi Annan, the former UN secretary-general.
Both sides have agreed to set up a committee that will outline policies for the coalition government to be formed in the near future.
The newly formed cabinet coalition will replace the previous government announced by Kibaki, days after he was controversially declared the winner of the presidential polls.
Last month's accord had been well received in Kenya but tough negotiations lie ahead for both sides as the government line-up remains yet to be decided.