The German Cup winners, who also beat Bayern Munich for the German Super Cup two weeks ago, looked to be cruising to an easy win after Ivan Perisic wrong-footed the Eintracht keeper with a clever header at the far post in the 13th minute and Bas Dost volleyed in from a deep throw-in four minutes later.
Eintracht, however, quickly pulled one back, when Stefan Reinartz, their new signing from Bayer Leverkusen, was left unmarked inside the box and had no trouble drilling in a header two minutes later.
Midfielder Kevin De Bruyne, whose future at the Wolves is in doubt with Manchester City interested in the Belgium international, was kept in check until his second-half substitution and Eintracht went close to an equaliser when substitute teenager Joel Gerezgiher’s powerful shot was parried by Koen Casteels.
Cologne scored twice in two minutes against the run of play to beat hosts VfB Stuttgart 3-1. Stuttgart, who narrowly avoided relegation last season, were by far the better side and missed a hatful of chances until keeper Przemyslaw Tyton brought down Anthony Modeste, who converted his 75th-minute spot-kick.
Before Stuttgart had time to regroup Cologne struck again through substitute Simon Zoller.
The hosts hit back two minutes later with a penalty from Daniel Didavi, but a stoppage-time goal from Yuya Osako sealed an unexpected win for the visitors.
Champions Bayern Munich shot out of the blocks, crushing hapless Hamburg SV 5-0 on Friday while Borussia Dortmund outclassed Borussia Moenchengladbach, who finished third last season, 4-0 with two goals from Henrikh Mkhitaryan.
(Reporting by Karolos Grohmann, editing by Ed Osmond)