In a game of few chances, substitute Borges slid home the only goal of the game on 83 minutes after a fine cross from fellow second-half sub Luca Barrington.
The victory means City finish third in Group A, three points adrift of second-placed PSG.
Leipzig finish bottom with six losses.
City, who beat RB Leipzig 5-1 in the season's first UYL group stage meeting in Manchester back in September, had few chances in an even if uneventful first 45.
Oscar Bobb's fine jinking run as he cut in from the right saw him drift past several challenges only to fire a weak shot that failed to trouble Fabian Eutinger in the hosts' goal.
Other than that, under a dull grey sky at a chilly Stadion am Cottaweg, there was little else of note to report in the first-half of a game where only pride was at stake for both teams.
After the break, Brian Barry-Murphy's youngsters had more possession and more intent.
Though the home keeper was rarely troubled, the threat around the box was continuous as City looked for the break-through.
The introduction of Barrington and Borges, however, would prove inspired.
Both were brought on after 61 minutes and the pair combined to create a winning goal that ends our European campaign for this season on a positive note.
Barrington. out on the left of the Leipzig box, slid a low ball across the face of the goal and Borges' outstretched left foot diverted the ball home to secure victory.
What it means...
Sadly, City could not progress in the UEFA Youth League regardless of the result in Leipzig.
Eight points from six games secures third spot for the Under-19s.
Brian Barry-Murphy's side are not in action again for 10 days.
The EDS will return to PL2 action when we take on Brighton & Hove Albion at the Academy Stadium on Friday, 17 December, kick-off 7pm (UK).
City: Slicker, Lewis (Susoho 61), Burns, Mbete (C), Taylor, Charles, Breckin (Adam 90), O'Reilly, Bobb, Ogwuru (Borges 61), Kayky (Barrington 61)
Subs: van Sas, Oduroh, Galvez
News about Leipzig v Man City"
Zinchenko: Every player is primed to show his best
Oleks Zinchenko says every player in the City squad is giving their all in the hope of selection. Read more
Guardiola: Changes possible for Leipzig
Pep Guardiola says he may ring the changes for Tuesday night’s Champions League trip to RB Leipzig. Read more
RB Leipzig U19s v City EDS: Full-match replay
Relive every kick of our UEFA Youth League clash away to RB Leipzig Under-19s with a CITY+ subscription. Watch more
Zinchenko: We won't take our foot off the gas
Oleksandr Zinchenko asserts City have no intention of taking our foot off the gas in Tuesday night's Champions League trip to RB Leipzig. Watch more