Thibaut Courtois made his Chelsea debut and Didier Drogba returned to the fold, but they couldn't help the Blues avoid defeat.
The match marked Thibaut Courtois' first outing in a Chelsea shirt, having returned from a three-season loan spell at Atletico Madrid. But the goalkeeper's debut did not go according to plan, with Elia scoring from the penalty spot and Obraniak doubling Bremen's advantage before halftime.
Courtois was replaced by Petr Cech at the break, while Chelsea legend Didier Drogba also made his second debut for the club having signed a one-year deal last week.
However, the Ivorian was unable to spark a Chelsea revival and the London club fell further behind when Kroos converted a late penalty after Nils Petersen was brought down by Nathan Ake.
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Courtois was called into action in the eighth minute when former Chelsea man Franco Di Santo headed goalward. The 22-year-old palmed that effort away.
But captain John Terry was guilty of using his own arms to block a cross 20 minutes in, opening the door for Elia to slot beyond the Chelsea debutant from 12 yards.
Jose Mourinho's men attempted to push for an immediate response, but appeared short of ideas going forward, with Bremen able to deal with everything the visitors mustered.
And Courtois was left to pick the ball out of the net once again eight minutes before the break, when Obraniak rose well to direct a header in the bottom left-hand corner.
Mohamed Salah found the net for the visitors before the break, but he was ruled offside following good work from Chelsea newcomers Cesc Fabregas and Diego Costa.
Mourinho introduced Drogba in a bid to change his side's fortunes, but the sheer number of changes that followed robbed the match of any flow or tempo, meaning chances were at a premium.
The match threatened to liven up once again in the closing stages, and Kroos' 89th-minute penalty put the result beyond doubt.