Wales, who won the trophy four years ago when they beat Argentina in the final, will also face Fiji's Pacific neighbours Tonga and outsiders Uruguay in Group E following an elaborate draw ceremony held at the Petroff Palace in Moscow.
The sport is being included on the Olympic programme for the first time, starting from 2016.
Twenty four teams were drawn in six first-round groups, with the top team from each group, together with the two best runners-up progressing to the quarter-finals.
New Zealand, ranked number one in the world in the shorter as well as the longer version of the game, were drawn to meet North American rivals, Canada and the United States, as well as Georgia in Group D.
Australia, France, Spain and Tunisia comprise Group A, while South Africa, Scotland and Japan will face hosts Russia in Group B.
African nations Kenya and Zimbabwe were placed with Samoa and Philippines in Group C, while 1993 champions England will take on Argentina, Portugal and Hong Kong in Group F.
Kenya are an emerging force on the sevens circuit, earlier this month beating New Zealand in a huge shock and then leading England with only seconds remaining in the NZ Sevens final before losing in sudden death extra time.
"There are no easy groups," International Rugby Board (IRB) chairman Bernard Lapasset told a news conference.
"I think that all the groups are evenly balanced which should make for a very exciting competition."
The 16 women's teams are also taking part, with Australia, who won the inaugural women's World Cup in 2009, facing South Africa, Ireland and China in their first-round group.
Lapasset said he was hoping Sevens will finally be able to come out of a shadow from the traditional 15-a-side version.
"It's a big issue right now," the Frenchman told reporters.
"The 15 is a different sport, a more traditional, while the sevens is a fast-paced very attractive game which is becoming more and more popular around the world.
"This is the last World Cup before the Olympics so I'm sure most of the world's best players from all rugby codes will come to Moscow to participate," he said.
"I also hope the event will be well attended. Rugby is becoming more and more popular in Russia," added Lapasset, calling Russia "an up-and-coming force in rugby".
The IRB chief assured the competing nations that visa restrictions will be eased off for the June 28-30 tournament. The final will be staged at the 80,000-seat Luzhniki Olympic stadium.