Daniel Batty got the faintest of touches to a Kamil Grosicki shot to edge Hull ahead after only three minutes, but Borja returned from obscurity to spark the Swansea comeback.
The Spanish striker scored his first Swansea goal for 1,022 days when he headed the equaliser, before Mike van der Hoorn followed suit just one minute and 47 seconds later.
Jarrod Bowen started for Hull despite interest from Premier League clubs while Cooper handed Borja a spot 950 days since his last start.
Both Cooper and Grant McCann were eager to impress in their first competitive games in charge, but it was the Hull manager’s reign that got off to a dream start after just three minutes.
Midfielder Jackson Irvine deceived the defence with a disguised flick to give Grosicki space on the edge of the box, and he used his time to send a curling shot towards goal.
Swansea debutant Freddie Woodman dived down to his left to make the save, but Batty got a toe to the ball to divert it into the bottom corner for his first goal for the club.
The hosts had an opportunity to level 10 minutes later after Borja’s header was scrambled clear to Jay Fulton, but his follow-up shot failed to get through a crowd of bodies.
Hull held out to take their slender advantage into half-time, despite only having 31 per cent of the ball, but Swansea turned the tables with two goals in as many minutes after the restart.
Almost three years on from his last outing in a Swansea shirt, Borja showed his poacher’s instinct to lift a header over George Long and into the net after Fulton had flicked Nathan Dyer’s cross into the air on the six-yard box.
The Swansea supporters were on their feet again in the 49th minute when Van der Hoorn stooped to head them in-front. Jake Bidwell had seen his cross clip the crossbar and fall to Dyer, who hit the ball back into the ground for the Dutch centre-back to dive in and head home.
The Swansea assault continued after a mix-up in the defence presented Dyer with an open goal, but Long scrambled back to deny the busy winger.
Hull went in search of a late equaliser when Irvine got on the end of a Batty corner and sent a header just over the bar.
Cooper’s side looked nervous in the final minutes but held on to get off to a perfect start under their new manager.