But Serie A leaders Lazio maintained their good form by beating Empoli 2-0.
Inter coach Hector Cuper fielded a mix of reserve and youth players at Bari and he will probably be content to only have a single goal deficit to overcome in the second-leg at home in two weeks time.
Bari's goal came in the 25th minute when Gaetano D'Agostino headed home a cross from Mattia Collauto.
In the second half Inter introduced a familiar name into their little-known line-up when Mattia Altobelli, 19-year-old son of former Inter and Italy striker Alessandro, made his first-team debut.
Roma also handed some fringe-players a rare start in Trieste against the surprise side of the Serie B season.
Triestina are second in the table and showed why as they put Roma under plenty of early pressure.
Roma goalkeeper Ivan Pelizzoli did well to keep out a thundering free-kick from Alessandro Parisi in the 20th minute and the home side enjoyed the
Triestina took the lead in the 63rd minute when Roma's Greek defender Traianos Dellas brought down Mirco Gubellini for a penalty.
Matteo Beretta's spot-kick was parried by Pelizzoli but Beretta followed in on the loose ball and fired into the net.
It looked like it would be a memorable win for the north-eastern side until four minutes from the end when Batistuta's low free-kick from over 20 metres out skidded through a crowded area and into the bottom corner.
Roberto Mancini's Lazio, unbeaten since the opening day of the season in any competition, took the lead in the 10th minute when defender Giuseppe Pancaro fired home after a goalmouth scramble from a corner.
Despite leaving out most of their first team Lazio were never in danger and doubled their lead seven minutes after the break when Simone Inzaghi latched onto a poor backpass from Brazilian defender Cribari and was brought down by goalkeeper Gianluca Berti.
Inzaghi converted the penalty to give Lazio some breathing space for the second leg in two weeks' time.
League champions and last season's Cup runners-up Juventus face Reggina in Thursday's games.