Rodriguez was back in action for the first time since mid-October, as he took a big step in his recovery from foot surgery.
The striker started at Hartlepool’s Victoria Park stadium, going close with a couple of chances before being withdrawn in the 61st minute.
While his return was one major positive, the other was Saints ending a run of eight matches without a win in the Barclays Under-21 Premier League.
They had to come from behind as well, after Arnel Jakupovic put the hosts into an early lead, but Josh Sims netted a first-half equaliser on his return to the side, before Sam Gallagher scored twice after the interval to win it.
Technical Director Martin Hunter made three changes from the side that lost 2-1 at Leicester City last week, with Rodriguez being joined by the returning pair of Dom Gape and Sims.
Matt Targett, who played for the first-team in Saturday’s 2-1 win at Stoke City, was not involved, while Richard Bakary and Olufela Olomola dropped to the bench.
Saints started well and went close in the ninth minute when Sims, back in the side after being a substitute in the previous three games, saw a shot inside the area well blocked by a sliding Callum Johnson.
Rodriguez then went close in the 12th minute with an angled shot from the corner of the area that Boro keeper Joseph Fryer tipped away, before he held a low shot from Sims seconds later.
Saints then fell behind in the 16th minute, as Jakupovic converted from close range after Paulo Gazzaniga had saved Jordan Jones’s curling effort towards the far corner.
Jakupovic then tested Gazzaniga with a free-kick from 25 yards, but the Saints keeper was equal to it, tipping it over.
The visitors then got themselves back into the game, as Harry Reed worked the ball wide to Gape on the right, and the captain’s superb low delivery was met at the back post by Sims, who side-footed into the roof of the net.
Rodriguez then had a good opportunity from a central position 18 yards out, but he hit a first-time, side-footed shot from Jake Hesketh's left-wing delivery a yard or so wide of the post.
Like Rodriguez, Sims continued to prove a threat and, shortly after half-time, he burst forward down the inside left and did well to strike a low shot under pressure that Fryer got down to hold.
Saints again went close to taking the lead when Rodriguez got free in the right side of the area in the 56th minute, but his low shot at the near post was kept out by Fryer.
Rodriguez was withdrawn in the 61st minute for Olomola, before Saints’ pressure finally told seven minutes later, Sam Gallagher raced through onto Hesketh’s pass and confidently finishing past Fryer with his right foot.
It was no less than Saints deserved, although Alfie Jones had to throw himself bravely in front of a Jakupovic shot soon after to keep the advantage intact.
There was a sour note at the start of added time, after Gape was carried off with what looked like a bad injury following a nasty clash with Mark Kitching.
There was still time after that for Gallagher to round off the day, breaking through to finish with a low right-footed shot for Saints' third.
Middlesbrough Under-21s (4-4-2): Joseph Fryer; Callum Johnson, Rhys Williams, Matthew Elsdon, Mark Kitching; Thomas McAloon (Callum Cooke 61), Alex Pattison, Robbie Tinkler (c), Jordan Jones (Nathan McGinley 82); Arnel Jakupovic, Joao Morelli (Lewis Maloney 76). Unused substitutes: Luke Coddington (GK), Josef Wheatley.
Booked: Callum Johnson.
Goal: Jakupovic (16).
Southampton Under-21s (4-1-4-1): Paulo Gazzaniga; Dom Gape (c) (Yan Valery 90+1), Ollie Cook, Alfie Jones, Jake Vokins; Harry Reed; Sam Gallagher, Jake Hesketh, Armani Little, Josh Sims; Jay Rodriguez (Olufela Olomola 61). Unused substitutes: Harry Lewis (GK), Richard Bakary, Tyreke Johnson.
Goals: Sims (31), Gallagher (68, 90+7).
Referee: Billy Khatib.