Saturday's match was marked by Dortmund fans' protests against their club's sponsorship deal with a German arms manufacturer.
Frankfurt's Fares Chaibi had a big chance in the 71st minute but was unable to react quickly enough to turn Niels Nkounkou's fiercely-struck cross into the goal from close range.
Then Gittens made more of a harder chance at the other end a minute later, eluding a defender on the left of the penalty box before firing in from a narrow angle.
The 20-year-old English winger then sealed the win on a counter-attack in the 93rd minute.
Bundesliga runners-up Stuttgart started the new season with a 3-1 loss at Freiburg.
Ermedin Demirovic fired Stuttgart into an early lead at his former club with a scissors-kick in the second minute, but that was as good as it good for Sebastian Hoeness's team, which amassed a club-record points total last season.
Freiburg, playing their first league game since long-serving coach Christian Streich's retirement, responded with three goals to give new coach Julian Schuster a winning start.
Leipzig survived a sending off to beat Bochum 1-0.
Bochum were unable to capitalise on Leipzig captain Willi Orban's late sending off for denying Myron Boadu a clear goal chance. The 19-year-old Antonio Nusa's second-half strike on his Bundesliga debut was enough for Leipzig.
Holstein Kiel endured a tough welcome to the league in a 3-2 loss at Hoffenheim, where the promoted side conceded an early penalty and had a player sent off late.
Augsburg drew with Werder Bremen 2-2, while Mainz held Union Berlin 1-1.
