The 38-year-old Klitschko, who holds the WBA, IBF, WBO and IBO crowns, has now been unbeaten for 10 years with his most recent defeat back in April 2004 against Lamon Brewster.
Klitschko, with his brother Vitaly - the former WBC champion and a politician in his home country - in his corner, knocked down his opponent after a few seconds in the first round, a big left-right combination sending the 34-year-old Leapai sprawling.
The Ukrainian, whose record now stands at 62 wins with 52 knockouts and three defeats, then pummelled the hapless Australian in round five, knocking him down once again before a clean knockout with less than a minute left in the round.
"Alex Leapai, I know it was not easy for you but you were up against the best fighter in the world and it was hard," Klitschko said in a ringside interview.
"Despite my 38 years I felt superb. I was in top form tonight in the ring."
Klitschko also had a special mention for his home nation, saying his preparation for the fight was not easy due to the political crisis back home with neighbouring Russia.
"I trained long and hard but it was not easy. My mind was back in Ukraine and what is happening there. Politicians must find a compromise and it should not come to a war. Ukraine will not be split up and (must) remain united."
Leapai, who got a shot at the world title after his stunning win over Russian Denis Boytsov in November last year, never got much chance to fight, with the 15-centimetre taller Klitschko tormenting him with his jab and longer reach.
"I tried to go inside and avoid the jab but it did not work. It's like that when you fight the champion," Leapai said. "Now it's back to the gym. This is not the end of the road and I will come back."
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- Alex Leapai