Rogic, who has been in brilliant form at Celtic under Ange Postecoglou, scored one goal and teed up another for Melbourne local Jamie Maclaren.
Craig Goodwin and Riley McGree scored their first international goals off the bench in the second half to ensure Australia snapped a three-game winless run to remain in touch with leaders Saudi Arabia and Japan in Group B.
Japan kept hold of second place in the group, a point ahead of the Socceroos, with a 2-0 win over China. The nations meet on March 24 in Australia, in what could prove to be a decisive tie for one of the two automatic qualification places.
Assistant coach Rene Meulensteen took the reins on Thursday in Arnold's COVID-19 enforced absence.
Arnold had been due to finish his quarantine period on Thursday but did not clear the required protocols to to attend. A Football Australia spokesman said he would be assessed daily as to whether he could travel to Oman for the qualifier on Feb. 2.
The change of voices on the sideline seemingly did not impact Rogic, who was involved in everything in an electrifying first half, and looked to have rifled Australia into the lead inside 20 seconds, sending the AAMI Park crowd of 27,740 into raptures.
But after a VAR referral, referee Ko Hyung-jin ruled an offside, determining that Jackson Irvine had interfered with Vietnam goalkeeper Bui Tan Truong.
From there, Australia's creative moments were few and far between until the commanding Rogic helped deliver the breakthrough on the half-hour mark.
The playmaker lofted a cross towards Maclaren, who on his first start since June, slipped unmarked between two defenders to power home a bouncing header.
In stoppage time, Mathew Leckie slipped a pass through to Rogic who coolly double Australia's lead.
To the delight of the large and vocal Vietnamese contingent, the visitors burst out of the break, with Pham Tuan Hai lashing a shot just wide minutes after the restart.
Ryan made a reflex save to deny Nguyen Cong Phuong in the 62nd, then four minutes later, Nguyen Thanh Chung headed a close-range effort over the bar.
Ryan then turned provider, hoofing a long ball into the path of Goodwin, who, six minutes after entering the fray, coolly chipped Bui Tan Truong to settle Australia's nerves.
Shortly after, Goodwin teed up McGree for the young gun to cut onto his right and lash home, rounding out a successful first Socceroos game in Melbourne since December 2017.
With Aziz Behich unavailable, Sydney FC youngster and Olyroo Joel King started at left-back in his first cap, while Melbourne City sensation Marco Tilio received a late debut off the bench.