Carlton Morris headed Shrewsbury to Wembley for the first time since 2009 with the only goal of the game against Yeovil in their EFL Trophy semi-final.
Morris powered in Shaun Whalley’s cross to book Paul Hurst’s side a place in the final against Lincoln on 8 April.
The result continues a superb season for Shrewsbury, who are also chasing promotion to the Championship.
The Glovers almost forced extra-time when Otis Khan went through on goal in stoppage time, only to be denied by a strong left hand from Shrewsbury goalkeeper Dean Henderson.
Shrewsbury manager Paul Hurst told BBC Sport “To get to any final is a great achievement.”
“It gives everyone a day out, although from a playing point of view, you’re going there to try and win a game of football – you don’t want to go there just to say we played at Wembley.”
“I didn’t expect an easy game and we didn’t get one. We had lots of possession, we looked the better side but couldn’t work their keeper enough, certainly first half.”
“I’m just delighted, and really pleased with the crowd. There were a lot more in the ground than I thought, and they created an atmosphere.”
Yeovil boss Darren Way was full of praise for Shrewsbury but was disappointed to be knocked out of the cup.
“They’re a good team, they’ve got strength, pace, power all over the pitch and can cause you problems.
“But we’re very disappointed to have conceded from a set-play, and to be one game from Wembley and lose it in that manner is disappointing.
“There’s no time to dwell on it, it’s been a good cup run to the semi-finals and my team gave everything towards the end.
“Overall, I have to say well done to the players for getting this far, but we have a big end to the season now and I have to make sure they are up for the challenge.”