Morneau made the announcement after a meeting between Canadian prime minister Justin Trudeau and Chinese premier Ki Keqiang.
The Chinese-led AIIB was announced in Beijing in 2014, with the aim of boosting investment in infrastructure in Asia and was established formally by the 57 member countries signing the Articles of Agreement in June 2015. The US has not joined the bank.
"Succeeding in the global economy of tomorrow will require strategic partnerships and openness to the world. That’s why, in addition to helping families and making historic infrastructure investments at home, we are forging new ties with international partners. Canada is always looking for ways to create hope and opportunity for our middle class as well as for people around the world, and membership in the AIIB is an opportunity to do just that," Morneau said in a statement.
AIIB president Jin Liqun said: "The decision of Canada to apply to join AIIB is very welcome and shows its confidence in the strong foundations the bank has built in our first few months. Canada has a remarkable track record of multilateral engagement, and I believe it will contribute significantly to the development of this bank."
Canada will become a member "early in 2017", the AIIB said.
The AIIB is expected to have initial capital of close to US$100 billion to invest in infrastructure projects and will be financed by individual country contributions proportionate to their economic size.