Edwards played the full game for Partick while Rogic came off the bench just after the hour mark for visitors Celtic, who took the lead in the 24th minute when midfielder Olivier Ntcham fired in a drive which took a deflection off Hoops skipper Scott Brown.
The treble winners, with a UEFA Champions League play-off tie against Astana coming up next week, dominated for large spells but found creating chances increasingly more difficult, giving the battling Jags hope all the way to the final whistle.
After starting with five teenagers in the convincing 5-0 Scottish League Cup win over Kilmarnock in midweek, boss Brendan Rodgers brought back experience with his eight changes.
James Forrest replaced benched striker Leigh Griffiths as the main attacker with Brown, Mikael Lustig, Nir Bitton, Jozo Simunovic, Ntcham, Callum McGregor and Scott Sinclair also coming into the starting line-up.
Partick signed strikers Miles Storey and Conor Sammon earlier in the day and the latter, on a season-long loan from Heart of Midlothian, started up front while Storey, who arrived from Aberdeen on a two-year deal for an undisclosed fee, was on the bench.
Thistle, skippered by Kris Doolan, were soon chasing the ball around the pitch as Celtic quickly got into their passing game.
In the ninth minute Jags keeper Tomas Cerny made a decent save from Forrest's drive on the turn inside the box, after Sinclair had sliced through the Firhill defence with ease.
The breakthrough came after Cerny made a fine save from Bitton's curling free-kick from 25 yards out at the expense of a corner.
Winger Jonny Hayes's cross was cleared but only to Ntcham and the 21-year-old midfielder, signed from Manchester City in the northern summer, fired it straight back in with the ball appearing to come off Brown before it ended up behind Cerny.
It was all too comfortable for the champions at that point and they came close to a second when Brown headed a Lustig cross on to the roof of the net.
Six minutes after the break Kieran Tierney robbed Steven Lawless and drove into the Thistle penalty area only to drag his shot wide of the far post, seconds before Doolan headed a Christie Elliott cross over the bar when he should have scored.
Storey replaced Doolan and, before Chris Erskine came on for Lawless, Rodgers replaced McGregor and Hayes with Griffiths and Rogic.
That double substitution energised the Parkhead men who would have made it two in the 70th minute had Callum Booth not headed Simunovic's header from a Griffiths corner off the line.
Thistle became emboldened in the nervy final stages, which gave Celtic opportunities on the break, but Cerny was largely untroubled and there was a late and unsuccessful penalty shout from the home fans for a Bitton tackle on Storey.