The reforms - which would significantly increase the powers of President Recep Tayyip Erdogan - received the required three-fifth majority in the assembly.
The bill will be put to a referendum, probably in April, and will become law if more than 50% of voters back it.
Critics say the move could usher in authoritarian rule, and amounts to a power grab by Mr Erdogan.
But the president says the changed system will resemble those in France and the US.
The constitutional reform bill was approved by 339 of the 550-member assembly.
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Mr Erdogan assumed the presidency, a largely ceremonial position, in 2014 after over a decade as Turkey's prime minister.
The reforms would let the president retain ties to a political party, so Mr Erdogan could resume leadership of the AK Party (AKP), which he co-founded.
Opposition leaders say that would abolish any chance of presidential impartiality.
It has been suggested that Mr Erdogan could remain in power until 2029 under the new rules.
The draft constitution states that the next presidential and parliamentary elections will be held on November 3, 2019.
The president would have a five-year tenure, for a maximum of two terms.
If the clock on Mr Erdogan's presidency re-sets from 2019, that would allow him to stay in the job until 2029, not 2024.
Mr Erdogan has not made clear whether this would be the case.
Turkey's Prime Minister Binali Yildirim congratulated MPs on passing the constitutional changes, saying: "We have done our job. Now we convey the issue to its real owner, our people."
Kemal Kilicdaroglu, the leader of the opposition Republican People's Party (CHP), urged Turks to "spoil the game that was played in parliament".
He described the approval of the presidential system as a "mistake".
Debates over the constitution changes have been heated.
Last week a fight broke out in parliament after the AKP clashed with CHP members when an MP tried to film a voting session during a debate.
The pro-Kurdish Peoples' Democratic Party (HDP) boycotted the vote.
To secure its necessary majority, the AKP has relied on the support of the rightwing Nationalist Movement Party, the fourth-largest in the legislature.
Turkey has been in a state of emergency since a failed coup in July. The status was extended after a series of attacks on the country, including a mass shooting in an Istanbul nightclub on New Year's Eve.

Beatrice Lovane, 22, from Rochdale, fell ill before 21:30 BST on 26 August.
Returning a narrative verdict Heywood coroner Lisa Hashmi said she died of liver disease and damage caused by painkillers.
North West Ambulance Service admitted errors and is staging another inquiry.
Miss Lovane who was born in Harare, Zimbabwe, died in Fairfield General Hospital, Bury almost five hours after emergency services were called.
At Rochdale Coroner's Court, Ms Hashmi said "timely steps were not taken to instigate emergency treatment and basic life support" and there were "delays in indentifying the cause of her sudden collapse".
However, Ms Hashmi added it was "not possible to link the gross failure in care to the cause of death".
The North West Ambulance Service (NWAS) NHS Trust told the inquest, which concluded on Friday, it had identified a number of significant errors and omissions by the crew who attended.
After the verdict a NWAS spokesperson said in a statement it had received a complaint from Miss Lovane's family in March and a "full and thorough" investigation was undertaken.
The spokesperson added: "New evidence has recently been disclosed and as a result of this two paramedics and an emergency medical technician have been suspended from duty pending further investigation.
"We fully accept that the level of care Miss Lovane received was below the standard we would expect."
Turkey's parliament has approved a new draft constitution paving the way for a presidential system of government.
[NEXT_CONCEPT]
Three ambulance crew members have been suspended after an inquest heard there were delays and "gross failure" to provide care to a woman who died from an adverse reaction to paracetamol.