And their task became even harder when Klose pounced in the opening stages to put the run-away Group D leaders ahead.
Koumas, who manager John Toshack had identified as his main playmaker now Ryan Giggs has retired, lost the ball inside his own half, giving striker Kevin Kuranyi a clear run at the retreating Wales defence.
His pass was perfectly weighted into Klose's path and the captain's first-time strike with the outside of his right foot found the bottom corner.
Germany began to seize complete control and the home side need a superb last-ditch lunge by Lewin Nyatanga to stop Kuranyi from doubling the lead on 23 minutes.
The tall striker had been put through by a deft touch by Bastian Schweinsteiger, who pulled the strings all evening in Germany's midfield.
Kuranyi was again denied as the half drew to a close when James Collins blocked his shot inside the box after Wales were again guilty of giving the ball away cheaply.
Lone striker Freddy Eastwood endured a difficult first-half on his home debut, but the Wolves striker finally got his first sight on goal just before the break when his far-post header was blocked for a corner.
Robert Earnshaw was sent on for the second half to give Eastwood more support, and the change seemed to give Wales more attacking impetus.
Germany, though, slowly started to reassert their authority and Wales goalkeeper Wayne Hennessey was called into action twice in quick succession.
The excellent Schweinsteiger then shaved the crossbar with a beautiful curling effort, but the game was over as a contest moments later when Klose struck again.
Bale was the culprit this time as he was robbed on the right flank by Roberto Hilbert, and his perfect chipped cross was headed in by Klose.
The German captain then had two excellent chances to complete his hat-tick.
He first fired straight at Hennessey, who again showed composure beyond his 20 years, and then sent a free header wide of the post.
Wales kept pressing until the final whistle, but their limited attacking ideas were easily snuffed out, leaving Toshack with a tough task to lift his players before Wednesday's visit to Slovakia.
Wales:
Hennessey, Ricketts, Gabbidon, James Collins, Nyatanga, Ledley (Earnshaw 46), Bale, Koumas (Fletcher 67), Robinson, Davies (Crofts 79), Eastwood.
Subs Not Used: Price, Morgan, Llewellyn, Jones.
Booked:
Gabbidon, James Collins.
Germany:
Lehmann, Arne Friedrich, Mertesacker, Metzelder, Pander (Trochowski 46), Hilbert, Hitzlsperger, Schweinsteiger, Jansen, Kuranyi (Podolski 72), Klose (Helmes 87).
Subs Not Used: Hildebrand, Odonkor, Manuel Friedrich, Rolfes.
Goals:
Klose 5, 60.
Att:
25,000.
Ref:
Manuel Enrique Mejuto Gonzalez (Spain).
