The club captain Lorenzo Pellegrini gave Roma the lead when he converted Henrikh Mkhitaryan’s cross.
Andreas Cornelius came off the bench to head in an equaliser for Trabzonspor.
But Uzbekistan striker Eldor Shomurodov poked home a late winner on his Roma debut following his summer move from Genoa.
Roma will qualify for the group stage of the new third-tier European competition unless they lose next Thursday’s second leg at Stadio Olimpico. Roma face Fiorentina in their Serie A opener on Sunday.
Former Chelsea, Manchester United and Tottenham boss Mourinho replaced Paulo Fonseca at the end of last season.