The full-back pulled the trigger from 25 yards, his effort taking a slight deflection on the way, leaving Jordan Archer no chance of getting back to the ball as it flew inside his near post.
Archer earlier made an outstanding save from the penalty spot as a keenly-contested game of few chances looked set to finish goalless at Southampton's Staplewood training facility.
However, after making another fine save from Sam McQueen, Archer was finally beaten by Targett nine minutes from time.
Harry Winks was selected for his Under-21 debut in the midfield three with Connor Ogilvie making his first start for the U21s at left-back. Will Miller later made his first U21 appearance off the bench.
Shaq Coulthirst was central to our best attacking moments and tested Saints' goalkeeper Cody Cropper early with a dipping volley from 20 yards.
Southampton were handed a golden chance to take the lead on 23 minutes when Kevin Stewart was penalised for handball from Targett's cross. Jake Sinclair stepped up and struck his spot-kick firmly to Archer's left but the Scottish U21 international got across to tip it onto the post and out to safety.
That gave the team a lift and we enjoyed our best spell for the rest of the first half.
Laste Dombaxe forced Cropper to turn away his free-kick from 30 yards and Giancarlo Gallifuoco met Winks' corner, only to head wide.
Cropper darted out to save at Kenny McEvoy's feet on 34 minutes and was tested again two minutes later when Coulthirst wriggled into the box from the left and let rip from the angle.
The final action of the first half saw Winks take a quick free-kick to Ruben Lameiras, who connected well from 20 yards only for Cropper to save again, this time down to his left.
The chances dried up in the second half and you always felt one goal would win it.
Soulymane Coulibaly curled just wide on 56 minutes, Harry Reed blasted over from distance and Coulthirst curled just wide on 73 minutes.
Archer made two saves in quick succession to deny Sinclair and McQueen, a smart one-on-one block on 75 minutes, but the key moment arrived five minutes later as Targett's free-kick zipped home off the wet surface.
There was time for Archer to make one more save from Reed in the final minute, with substitute Ryan Seager blasting the rebound over the bar.
Southampton: Cropper; McCarthy, Stephens, Turnbull, Targett; Sinclair, Gape, Reed, Isgrove; McQueen, Flannigan (Seager, 75). Unused subs: Young, Johns, Hesketh, Moore.
Spurs: Archer; McQueen, Gallifuoco, Stewart, Ogilvie; Dombaxe, Lameiras (McEneff, 76), Winks (Miller, 74); Coulthirst, Coulibaly (Oduwa, 62), McEvoy. Unused sub: Miles.