British MPs approved Prime Minister Theresa May's motion to ask the EU for a delay by 412 votes to 202.
Thursday's vote came a day after lawmakers rejected a no-deal Brexit under any circumstances.
May says Brexit could be delayed by three months - to the end of June - if MPs back her deal next week.
If not, then she warned of a longer extension, which may prompt Brexiteers within her own party to back her deal.
However, all of this is only possible if every EU member state approves the extension to Article 50.
EU leaders will meet in a summit next Thursday.
European Council President Donald Tusk said he will appeal to all EU members to be open to a long extension if the UK finds it necessary to rethink its Brexit strategy and build consensus around it.
The threat of a no-deal Brexit is also still a possibility and the government says it is preparing contingency plans for such a scenario.
MPs earlier rejected an attempt to secure another Brexit referendum by a significant margin.
Even campaigners for a second referendum said now was not the time to vote on it.
They also rejected a cross-party plan to allow parliament to take control of the Brexit process.
That motion was only rejected by a margin of two votes.
Ro Aram, Arirang News.