A place for veteran Hibernian forward Martin Boyle was among those cut.
Yengi, 25, had been loaned from Livingston to Japanese club Machida Zelvia in January amid his parent club's battle against relegation from the Scottish Premiership. He repaid Popovic's faith by scoring six goals in 22 appearances as his side finished third in Japan's East Region and reached the Asian Champions League final.
Volpato, 22, only received FIFA ratification of his switch from Italian football on Friday — the day before the squad was announced — after previously being eligible for Italy's under-21 side.
Former Brighton and Hove Albion goalkeeper Mat Ryan, now at Levante, and Melbourne City striker Mathew Leckie will equal the national record by attending a fourth World Cup, joining Tim Cahill in that exclusive group.
Former Dundee United left-back Aziz Behich, now at Melbourne City, and St Pauli midfielder Jackson Irvine are on track to feature at their third World Cup. Heart of Midlothian midfielder Cammy Devlin and Aberdeen-born Leicester City centre-back Harry Souttar — whose brother John is in the Scotland squad — are among those making their second tournament appearances. Swansea City centre-back Cameron Burgess, Watford winger Nestory Irankunda and Norwich City striker Mohamed Touré are also included.
Also missing from the final selection alongside Boyle are goalkeeper Joe Gauci, forward Brandon Borrello and former Hearts defender Kye Rowles, now with DC United, who had been added late to the training camp but did not make the cut.
Popovic acknowledged the difficulty of the final decisions. "Some difficult decisions had to be made — that's the nature of major tournaments. But it's also a credit to all the players involved over the past few weeks who worked extremely hard during an extended and challenging pre-camp."
Australia face Switzerland in a final pre-tournament friendly on Saturday before opening their World Cup campaign against Turkey on June 14. They are also in a group with the United States and Spain.
Australia World Cup squad:
Goalkeepers: Ryan (Levante), Izzo (Randers), Beach (Melbourne City).
Defenders: Behich (Melbourne City), Bos (Feyenoord), Burgess (Swansea City), Circati (Parma), Degenek (APOEL), Geria (Albirex Niigata), Herrington (Colorado Rapids), Italiano (Grazer AK), Souttar (Leicester City), Trewin (New York City FC).
Midfielders: Devlin (Heart of Midlothian), Hrustic (Heracles Almelo), Irvine (St Pauli), Metcalfe (St Pauli), Okon-Engstler (Sydney FC), O'Neill (New York City FC).
Forwards: Irankunda (Watford), Leckie (Melbourne City), Mabil (Castellón), Touré (Norwich City), Velupillay (Melbourne City), Volpato (Sassuolo), Yengi (Machida Zelvia).