The Switzerland international joined from Real Sociedad at the start of August and the striker showed he could be a valuable purchase with a brace at the Commerzbank-Arena.
Inter had gone ahead against the run of play through Ruben Botta's close-range finish, but Lucas Piazon - who joined Frankfurt on loan from Chelsea last month - soon levelled.
Seferovic then took centre stage with a six-minute first-half brace, showing good composure with a calm finish in the area before bagging his second from a rebound when Nelson Valdez's header had hit the crossbar.
The second half proved a lacklustre contest and Walter Mazzarri will be keen for his side to show more in Thursday's match at PAOK, with a UEFA Europa League play-off first leg against Stjarnan to follow a week later.
Handanovic was called into action again from the resulting corner, the Slovenia international making a fine point-blank stop from Martin Lanig's header.
Frankfurt's wastefulness was punished in the 25th minute as Inter stole ahead.
Inter's lead lasted less than a minute, however, the visitors were carved open as Piazon raced through on goal and, though his initial shot hit the post, he tapped home the rebound.
Frankfurt had a third five minutes before the break. Nelson Valdez headed Inui's cross against the woodwork but Seferovic was on hand to bundle home the rebound.
Both teams played the second half at a considerably lower tempo with little in terms of goalmouth action, Hernanes' 25-yard free-kick that cleared the bar the closest Inter came to reducing the deficit.
