Both are widely considered too expensive for the poor west African country’s budget.
President Macky Sall last month announced emergency legislation to scrap the Senate and turn over its annual budget of almost eight billion CFA francs (12 million euros, $16 million) to dealing with the impact of deadly floods.
At least 13 people have been killed in several weeks of flooding and thousands left homeless.
The post of vice-president, created in June 2009 by former president Abdoulaye Wade but never filled, was allocated an annual budget of more than two billion CFA francs (three million euros).
The institution was strongly criticised in June 2011 when Wade introduced a bill stating that if the president resigned, died or was unable to carry out his duties, he would be replaced by the vice-president.
Many of Wade’s critics claimed he was setting up a system for his son Karim, who held a ministerial post, to succeed him, but Wade denied having such “monarchical” plans.
He was defeated at the polls by Sall in March after 12 years in office.
The legislation, approved by the lower house on Wednesday, will shortly go before the 100-member Senate itself, before it is passed by Congress, which unites both houses, and then signed into law by Sall.