Star Australian playmaker, Rogic produced a bounce-back performance for the Hoops with his 80th-minute volley ensuring his side put an end to their run of back-to-back losses.
The Australian was everywhere early on for Celtic, having a shot saved after 12 minutes before creating a number of chances with his slick passing and driving runs at the opposition defence.
Leigh Griffiths opened the scoring to give the Brendan Rogers coached side the lead after 18 minutes but Partick found an equaliser midway through the second stanza to make for a nervy end for the visitors.
However, Partick's hopes of an upset came crumbling down soon after with Moussa Dembele and then Rogic scoring in quick succession to secure the win.
Daniel Arzani completed his move to Celtic Park on Friday night (AEST) but he was not considered for the Cup clash as fans anticipate his debut for the Scottish Champions.
Rogic wasn't the only Australian to find the back of the net in Scottish Cup action with Lyndon Dykes scoring for second division side Queen of the South in their clash with St Johnstone.
However, Dykes goal wasn't enough to help his side to victory as they went down 4-2 after extra time.
The Aussie duo of Ben Garuccio and Oliver Bozanic both saw action for Hearts as they snatched a 1-0 win against Dunfermline.
Olly Lee scored the winner for Hearts 11 minutes from time, with Garuccio playing 90 minutes at left back while Bozanic came on at the start of the second half.
Hearts' other Australian player Ryan Edwards was not in the squad for the match.
In England, it was a day of mixed results and one Socceroos midfielder Massimo Luongo would want to forget after his Queens Park Rangers side was torn apart in a 7-1 thrashing at the hands of West Bromwich Albion.
Luongo started the match, but probabaly wishes he didn't as the Baggies put six second-half goals past QPR after going into the break locked at 1-1.
The result has QPR anchored to the bottom of the Championship with three losses from as many games.
Jackson Irvine was also on the wrong side of a losing scoreline as his Hull City side went down 1-0 to Blackburn Rovers.
Irvine started the match and played the full 90 minutes for the Tigers in central midfield but was unable to help his side take a point from the contest.
It was better news for Socceroos captain Mile Jedinak who returned to Aston Villa's starting line-up in the centre of defence as his side held on for a 1-1 draw with Ipswich Town.
Ryan WIllams' Rotherham United couldn't stop Leeds United's incredible start to the Championship season under new boss Marcelo Bielsa as they suffered a 2-0 loss at Elland Road.
Williams played the full 90 minutes on the left wing after starting the match.
Frank Lampard's Derby County scored late but couldn;t get the better of Fringe Socceroos defender James Meredith and his MIllwall side as the Lion's held on for a 2-1 win at The Den.
Meredith started the match in his usual left back position and played the full 90 minutes.
Adam Federici was on the bench for Stoke as they snatched a 2-2 draw with Preston North End thanks to some last-minute heorics from cult icon Peter Crouch.
In League One, Kenneth Dougall played 90 minutes for Barnsley as the held AFC Wimbledon to a 0-0 draw, Ryan McGowan lasted the whole match as his Bradford City side was defeated 2-0 by Southend United and Bradden Inman was taken off after only 40 minutes in Rochdale's 2-2 draw with Fleetwood Town.
In League Two, the Harry Kewell-coached Crawley Town slumped to their third straight loss after they were defeated 1-0 by Port Vale.
Finally, Aziz Behich was a starter and played 90 minutes for his Turkish Super Lig club Bursaspor in their 0-0 draw with Kayserispor and Matt Jurman made his debut for Saudi Arabian club Al Ittihad, lasting the 90 minutes, as they lost 2-1 to Al Hilal.
