Morrison said he had spoken to the Tory leader to congratulate him and had committed to continuing the two countries' "very strong relationship".
Morrison vowed to move "very quickly, when the opportunity presents" on an "agreement between the United Kingdom and Australia on trade".
Any future bilateral trade deal would likely echo an Australia-EU trade deal that is still under negotiation, and Canberra may first want clarity on Britain's future trade relationship with the rest of Europe.
Morrison urged Johnson to resolve "some very difficult issues" around Britain's exit from the European Union in the interest of both parties and the "broader global economy".
Morrison said he hoped to meet Johnson at the G7 summit next month in Biarritz, France.