First-half goals from Chris Zebroski and Eunan O'Kane put the Gulls in charge ahead of the second leg at the Greenhous Meadow next Friday night.
Striker Zebroski opened the scoring with a neat finish in the 29th minute after good work from Gavin Tomlin, and O'Kane added a second on the stroke of half-time.
The midfielder hit a low shot from 20 yards and, despite his effort lacking power, the ball flew into the corner past Shrewsbury goalkeeper Benjamin Smith.
Paul Buckle's men will feel they have taken a significant step towards reaching the final at Old Trafford after an impressive, fully-deserved victory.
Shrewsbury were second-best throughout and face a big task to try and turn round the tie in the second leg on Friday.
Graham Turner's side created precious few genuine goalscoring opportunities but they threatened first when James Collins hit a 20-yard shot which Torquay stopper Scott Bevan held comfortably.
Yet the Gulls always looked more dangerous in attack and almost led in the seventh minute when Tomlin crossed for Billy Kee, who headed wide from just six yards out.
Zebroski was then afforded the time and space to hit a long-range piledriver from 30 yards which flashed just past Smith's left-hand post in the 15th minute.
Shrewsbury forward Matt Harrold then fired wide from 25 yards before the hosts were dealt a blow in the 25th minute.
Defender Ian Sharps succumbed to injury and was replaced by Niall Canavan.
James Collins then drove wide from 12 yards before the opening goal arrived in the 29th minute when Tomlin did superbly well to create space down the right flank.
He turned his marker and played a low cross into the feet of Zebroski, who took one touch to control the ball and another to rifle home a low shot from eight yards out.
Torquay almost claimed a second moments later as Jake Robinson's 20-yard strike was parried by Smith and, with the hosts in the ascendancy, they scored again in the third minute of added time at the end of the first half.
Jake Robinson, who is on loan at Plainmoor from Shrewsbury but is eligible to play against them in the play-offs, embarked on a strong run down the left flank.
He laid the ball back to O'Kane, who hit a low right-footed shot from 20 yards which Smith perhaps should have saved.
Kevin Nicholson almost added a third for the Gulls in the 59th minute but Smith made a smart stop and then also denied Tomlin to keep the score down ahead of the second leg at Shrewsbury next week.