Griffin suffered a hamstring injury during the first half of Wednesday's 19-14 defeat to the Pittsburgh Steelers, though he remained in the game until the fourth quarter of the contest.
McSorley eventually replaced him, going on to team up with Marquise Brown on a 70-yard touchdown pass that gave depleted Baltimore a late glimmer of hope.
It was one of two completed passes out of six attempts for the 25-year-old, who was a sixth-round pick in the 2019 NFL Draft.
Now McSorley is the only quarterback left on the 53-man roster ahead of the game with the Dallas Cowboys.
Starter Lamar Jackson remains on the Reserve/COVID-19 list having reportedly tested positive for coronavirus, though the reigning MVP could yet be available in time to feature next Tuesday.
Running back J.K. Dobbins, outside linebacker Pernell McPhee and offensive linemen Patrick Mekari and Matt Skura have all been activated from the list, however, boosting Baltimore as they bid to get their season back on track.
The Ravens have lost four of their previous five outings to own a 6-5 record, leaving them behind the unbeaten Steelers and the Cleveland Browns (8-3) in the AFC North.
Next up is Dallas, whose defense has given up an average of 382 yards to opposing teams this season, including a league-high 156.4 yards on the ground.
The contest, which had been postponed three times, came during a period in which the Ravens had at least one player test positive for 10 successive days.
They have also had to put 23 individuals on their reserve/COVID-19 list, either because they returned a positive test or had come into close contact with someone who contracted the virus.
Although that number was reduced to 17 for Wednesday's match, they were still forced to call on 10 players from their practice squad in order to fill their roster for the game, which saw the Steelers improve to 11-0.
The situation made the Ravens' preparations for the game far from ideal as their training facility had been closed for five days prior to Monday, with the team then limited to walk-throughs Monday and Tuesday.
Quarterback Robert Griffin III, who took the place of reigning MVP Lamar Jackson due to his positive coronavirus test, was the most vocal post-game, questioning whether the NFL had properly considered the players' safety as he also considered the impact of a lack of training.
"It's not about whether or not guys want to play," Griffin said. "It's about whether or not our safety is actually being taken into account. I can't say much more than that.
"I pulled a hamstring today. I've never pulled a hamstring in my life. You see guys going down left and right."
During the game, Griffin was seven of 12 for 33 yards and an interception, while he rushed 68 yards before being replaced by Trace McSorley as the Ravens (6-5) were left third in the AFC North.
But afterwards he was in little mood to discuss the game itself, as he also hit back at suggestions the team cannot be bothered to play, adding: "When people think, 'Oh, maybe they just don't want to play. They just don't want to do this.'
"It's not that -- we love football. We want to play football, but we also want to make sure our families are safe. Those things [family members becoming ill] don't get reported."
Ravens safety Chuck Clark chimed in as well, pondering why the players were allowed back into the training facility at all on Monday.
Clark said: "I don't know what comes with me saying this, but, of course, on Monday and Tuesday, we're wondering, 'Why were we allowed back in the building if we say everything is based off contact tracing and things like that, and that's what's told to us?'.
"We've got to look at some of those things."
The AFC North match-up was moved from Thanksgiving Day to the weekend after several Ravens players were placed on the reserve/COVID-19 list.
However, further cases followed, including a positive test for reigning NFL MVP Lamar Jackson, widely reported on Thursday.
Coach John Harbaugh then told ESPN the team would not be able to return to their training facility - shut down earlier in the week - until Monday at the earliest, putting the Steelers game in serious doubt.
And the second postponement of the game followed on Friday, also affecting next week's Thursday night fixture between Baltimore and the Los Angeles Rams.
Jackson would appear a major doubt to play in either match due to the recency of his positive test.
Back-up quarterback Robert Griffin III appeared to confirm Jackson had contracted the virus as he posted on social media.
Griffin wrote on Twitter on Thursday: "Praying for my brother @Lj_era8 and every player, staff member and their families dealing with COVID-19.
"Ensuring the safety of the entire organisation is important. Handling this outbreak within the team is bigger than football."
Jackson has had a more testing campaign in 2020, throwing for 15 TDs but six picks - already matching his total interceptions for 2019.
He has added three rushing scores, yet the 23-year-old has been sacked 24 times - one more than last year - for 138 yards and has had six fumbles.
However, the news still represents a blow to the Ravens, who can ill afford a third straight defeat after losing to the New England Patriots and the Tennessee Titans.
Baltimore have fallen to 6-4 and third in the division, a game behind the second-placed Cleveland Browns and four back of rivals the Steelers.
Heading into this game, Pittsburgh are the only remaining unbeaten team in the NFL, top of the AFC North with a 10-0 record.
Griffin has taken just 17 snaps across three games this season. Jackson had built a big lead on all three occasions, with Griffin completing only one pass this year and also throwing a pick.
Griffin's last regular season start came against the Steelers in 2019, when Baltimore ran out 28-10 winners but the QB threw 11-of-21 for no touchdowns and an interception.
