Boro went out to fellow League Two side Burton Albion at the equivalent stage of last year’s competition.
They lost 2-1 at the Pirelli Stadium thanks to two late goals from striker Shaun Harrod.
Now Boro captain Matthew Bates, who was on the losing side that day, is anxious to ensure there are no more slip-ups.
“We’re due a good cup run,” he declared.
“We haven’t had one for a few years and it was disappointing to go out at Burton Albion last year.
“It would be nice for the fans.
“I know supporters love the cup – especially the away games which are always a bit special and, to give the fans one of those, we have to get past this tricky one first and be in the draw for the next round.”
Boro, fourth in the Championship, are clear favourites against a team lying fifth in League Two.
“We are at home against a lower league side so it is a winnable game and, once you get past this stage, anything can happen,” added Bates.
“So long as we get a win, it doesn’t matter how we do it.
“I still get excited about the cup – all the players do. Everyone at the club does; the FA Cup is always a bit different.”
Boro are without midfielder Nicky Bailey, who could be out for up to two months with a knee injury that forced him to miss last week’s Championship defeat to Blackpool.
The 27-year-old damaged his medial ligament during a draw against Peterborough on New Year’s Eve.
“Nicky isn’t going to make it (the Shrewsbury game),” confirmed Boro manager Tony Mowbray.
“He will be out for anywhere between four and eight weeks – hopefully closer to four rather than eight. He’s done fantastically well for us this year and was evidently missed (against Blackpool) with the style of play he brings to the team.
“Most footballers come across an injury like this in their career, I know I did.”
Shrewsbury chief Graham Turner has defender Joe Jacobson available after suspension.