Ben Futcher celebrated his first win as interim coach after Rogers scored the opening goal from a penalty spot.
Austria's Thierno Ballo equalised for the visitors but second-half goals from Rogers, James McAtee and Tom Fellows put England in good stead to continue their 2025 Euro qualifying campaign.
Liam Delap picked up a very early yellow card inside 20 seconds for a dive in Austria's box, and it was the visitors who almost dealt the first blow.
England conceded a free-kick on the edge of the area which set up Austria's Christoph Lang, who chose to curl his effort around the wall, forcing goalkeeper James Beadle to tip his effort around a post to keep it level.
TrendingAustria's early warning shot kicked the hosts into gear, but a scuffed one-v-one attempt by Delap fired straight at the keeper was a reminder that England need to be more clinical should they wish to avoid a repeat of Friday's
goalless draw with Northern Ireland.
Futcher's men were well-warmed up by the 25th minute and their sharpness paid dividends when Tyler Morton caught Florian Micheler in possession.
The midfielder had his pocket picked after a pressure-inviting pass from his keeper and the loose lost ball found McAtee, who went through on goal before the Manchester City man forced the foul out of Nikolas Polster with quick footwork to win a penalty.
Aston Villa's Rogers, who had perhaps been unfortunate to have not been called up for Lee Carsley's senior squad, took on the responsibility, and he blasted his effort into the roof of the net to give England a 1-0 lead after 25 minutes.
But a mix-up at the back undid England's hard work as Austria took advantage of an error before a sublime long-range goal.
Skipper Taylor Harwood-Bellis fed a fizzed pass out from the back into the feet of Alex Scott. The usually-reliable midfield player mis-controlled his touch and was bullied to the ground as Ballo was set up nicely to curl his effort past the motionless Beadle into the bottom right corner for 1-1.
England came out of the traps quickly in the second half and they managed to retake the lead.
Samuel Iling-Junior performed a no-look pass into the feet of McAtee, who instinctively flicked Rogers into space. The winger took a single touch before he whipped his effort into the far right-hand corner in clinical fashion to grab his second and make it 2-1.
McAtee's dominant display went up a gear when he got in on the scoring act himself. Rogers was heavily involved, he jinked away from his marker and unleashed Lewis Hall on the overlap before his whipped low cross met McAtee on the volley and he steered his strike into the bottom right corner.
In the 81st minute, Hall replicated his earlier delivery, this time his cross met substitute Fellows who tapped in the fourth.