The National League South side have been drawn at home against National League opponents Aldershot for Saturday’s game.
Weston reached the first round last season, enjoying a draw against their then League One neighbours Bristol Rovers, who they took to extra time.
But the last time they got through to the second round was in the 2003-04 season when they were beaten by Northampton Town.
They come into the Aldershot tie second in the league table under Rogers – who stepped up from assistant to manager following the departure of Scott Bartlett to Eastleigh only 11 days ago – and in excellent form, with one defeat in 16 games stretching back to August.
“These are the games you can look back on and you tell people you played in the first round,” Weston defender James Pope said.
“On paper, we’re quietly confident, and [it] being a home tie is a good thing as well, we’re expecting a big crowd.”
Rogers, meanwhile, is hoping a win in the cup can help spark growth for the club.
“I played for Forest Green and Newport and you can see what they’ve gone on and done. Weston’s very similar, it’s got so much potential,” he said.
“We want to make the town proud of this football club, and having these cup runs the last two years can only enhance that.”
