Thomas Muller struck twice as world champions Germany stuttered against the determined Scots at Signal Iduna Park.
A 4-2 friendly defeat to Argentina on Wednesday raised questions in the Germany squad and, although Joachim Low's side were far from their best, a better display earned the hosts three points.
Scotland's joy lasted only four minutes, though, as Muller's scrappy second – scrambled in from five yards – was enough to defeat Gordon Strachan's plucky side, who saw Charlie Mulgrew sent off late on.
Bekim Balaj's superb volley claimed a shock 1-0 win for Albania in Portugal, with the home fans greeting the full-time whistle with intense jeers.
Gianni De Biasi's side were billed as huge underdogs against a team expected to cruise through Group I, but they left Aveiro with all three points as Balaj stole the show on a memorable day for Albania.
Balaj's winner arrived in the 52nd minute as he belted Odise Roshi's delivery past Rui Patricio, consigning Portugal to an embarrassing defeat in the absence of talisman Cristiano Ronaldo.
Grosicki doubled his tally and Lukasz Szukala added one of his own, but Robert Lewandowski stole the show by scoring four in the second half to complete Poland's biggest ever non-friendly win.
Also in Group D, Aiden McGeady's double gave Republic of Ireland the points as they began with a 2-1 triumph in Georgia.
Everton winger McGeady netted again, though, delivering an emphatic finish from the edge of the area to give Martin O'Neill his first competitive win as manager.
Claudio Ranieri's first competitive match in charge of Greece ended in despair as 10-man Romania took all three points away from Piraeus, winning 1-0 thanks to a 10th-minute penalty from Ciprian Marica, who was later sent off for two bookable offences.
