Forward Horta bagged his fifth goal of the season to snatch a 1-0 opening win in the Europa League Group K clash at Molineux.
Nuno Espirito Santo’s side won all their qualifying games to reach the group stage, including an impressive 5-3 aggregate victory over Torino, but are winless in the Premier League.
They go to Crystal Palace on Sunday with questions already being raised about their ability to cope in Europe and domestically.
With both teams struggling – Braga are third bottom of the Portuguese Primeira Liga – Europe offered a way to reignite their campaigns.
Yet it was no surprise the game failed to find an early spark and it took until the 21st minute to conjure the first real chance.
Raul Jimenez’s lobbed cross for Cutrone allowed the striker to hold off Bruno Viana but he only blazed over from 10 yards.
Before then only Ruben Neves’ hopeful effort and Horta’s blocked shot for Braga were worth a mention.
Braga had been kept at a distance by Wolves, with the returning Willy Boly excellent, but wanted a penalty after 31 minutes.
Galeno skipped past Conor Coady and went down looking for a spot kick but his pleas were correctly ignored by referee Jakob Kehlet.
Braga had been efficient enough but lacked courage in their convictions and Wolves went close a minute before the break.
Cutrone had shown enough endeavour but snatched at a rare opening when he collected Matt Doherty’s pass to fire wide from 16 yards.
It roused Wolves though and they began the second half with renewed vigour.
The hosts began to take control after Rui Patricio comfortably collected Paulinho’s weak effort and should have gone ahead after 51 minutes.
Jonny’s low cross deflected into Dendoncker’s path but Matheus saved his low strike with his legs.
Wolves continued to dominate but rather than finding a second-half breakthrough they were frustrated – before it got worse.
Braga’s threat had been minimal since the re-start but they silenced Molineux with the winner after 71 minutes.
It was a gift when Ryan Bennett’s loose pass gave the ball to Galeno on the halfway line and he raced clear.
With Wolves stretched the forward found the unmarked Horta and he drilled in at Patricio’s near post from 10 yards.
The unusually quiet Jimenez headed at Matheus with 11 minutes left and Boly’s low shot was gathered by the goalkeeper soon after.
Matheus then turned away Neves’ drive in stoppage time as Wolves failed to recover.