The original Arabic seems to be from Surah 54,1 Al-Qamar:

The Hour has come near, and the moon has split [in two].
__Translation into English from Quran.com

What that sentence alone might mean is of course up for debate, but the claim seems to explicitly refer to using the quote as is, albeit and in all likelihood in a German version; not the meaning or interpretation of it. That makes the first part quite mundane, and the second part with the description of attributes about the moon the significant part.

In German the Koran reads:

Näher ist die Stunde (des Gerichts) gekommen, und gespalten hat sich der Mond.

Or perhaps:

1. Die «Stunde» ist nah, und der Mond ist entzwei gespalten.

That is a sentence start that Hitler indeed loved.

Translated into German, that would render as Die Stunde naht/ ist nahe/ ist gekommen…

Of course, this is common expression in German. Yet, he had a special preference for it. And it seems equally insignificant as a special expression in the Koran or the Arabic language. To find older – and for a Catholic more fitting sources, one might consult the bible in John 12:23 or Matthew 26:45, and several more hits from the New Testament.

For that and close variations you will find many, many hits in his speeches:

Die Stunde der Auseinandersetzung mit den Novembermännern ist damit gekommen. / Die Stunde der Niederbrechung dieses Terrors kommt. / Es wird die Stunde kommen, da wir auch vor die Gräber der Gefallenen des großen Krieges hintreten und sagen werden können:…

Just to quote a few examples. As a Catholic he was also naturally prone to invoke religious imagery in his speeches, one of the better known examples:

Denn ich kann mich nicht lösen von dem Glauben an mein Volk, kann mich nicht lossagen von der Überzeugung, daß diese Nation wieder einst auferstehen wird, kann mich nicht entfernen von der Liebe zu diesem meinem Volk und hege felsenfest die Über­zeugung, daß eben doch einmal die Stunde kommt, in der die Millionen, die uns heute hassen, hinter uns stehen und mit uns dann begrüßen werden das gemeinsam geschaffene, mühsam erkämpfte, bitter erworbene neue deutsche Reich der Größe und der Ehre und der Kraft und der Herrlichkeit und der Gerechtigkeit. Amen."

__Translation: I cannot renounce the conviction that this nation will one day rise again, I cannot renounce my love for this people of mine and I firmly hold the conviction that the hour will come when the millions who hate us today will stand behind us and welcome with us the jointly created, laboriously fought, bitterly acquired new German kingdom of greatness and honour and power and glory and justice. Amen."

(February, 10, 1933)

This is however only the first part of the quote. And the examples I quoted were from way before Operation Barbarossa, or whatever we might choose to identify "when his armies rushed to Moscow."

He used that insignificant part of the quote indeed on June, 22, 1941, the day of the invasion into of the Soviet-Union (English version on Wikisource). In fact he opened his speech with it:

„Deutsches Volk! Nationalsozialisten!
Von schweren Sorgen bedrückt, zu monatelangem Schweigen verurteilt, ist nun die Stunde gekommen, in der ich endlich offen sprechen kann.

__Translation: German people! National Socialists!
Distressed by grave worries, condemned to months of silence, the time (hour) has come when I can finally speak openly.

And he continues in the same speech with it:

Damit aber ist nunmehr die Stunde gekommen, in der es notwendig wird, diesem Komplott der jüdisch-angelsächsischen Kriegsanstifter und der ebenso jüdischen Machthaber der bolschewistischen Moskauer Zentrale entgegenzutreten.

__Translation: Now, however, the time has come to oppose this plot by the Jewish Anglo-Saxon war instigators and the equally Jewish rulers of the Bolshevik Moscow headquarters.

But unlike the quote claims, this speech ends quite differently:

Die Aufgabe dieser Front ist daher nicht mehr der Schutz einzelner Länder, sondern die Sicherung Europas und damit die Rettung aller.
I habe mich deshalb heute entschlossen, das Schicksal und die Zukunft des Deutschen Reiches und unseres Volkes wieder in die Hand unserer Soldaten zu legen. Möge uns der Herrgott gerade in diesem Kampfe helfen!

__Translation: The task of this front is therefore no longer to protect individual countries, but to secure Europe and thus save everyone.
I have therefore decided today to put the fate and future of the German Reich and our people back into the hands of our soldiers.
May the Lord help us in this fight!

Quotes in German from: Max Domarus: "Hitler. Reden und Proklamationen 1932–1942. Kommentiert von einem deutschen Zeitgenossen.", Pamminger und Partner: Leonberg, 1988. (Availabe on archive.org[PDF-link])

Looking through all of these speeches searching for the occurence of "Mond" (moon) we find very few hits at all:

Ich bin auch kein Gymnasiast, der in einen Schulatlas Karten einzeichnet. Südamerika liegt so weit weg wie meinetwegen der Mond. Es sind das dümmste Behauptungen.
__Translation: I'm not a high school student who draws maps in a school atlas either. South America is as far away as the moon for me. These are the stupidest claims.

The background information surrounding this claim should raise doubts from the beginning. It just wouldn't make any sense to play with religious imagery from the Koran when soldiers are rushing to Moscow. It is known that Hitler had a "soft spot for Islam" whereas Himmler might have had even considerable more interest for that special book.
But the timing is quite off. So, if it were Himmler using that quote in a public speech when the Wehrmacht was rushing towards Belgrad, we would have seen some semblance of plausibility. After all the SS did recruit some Muslims into their ranks from that region.

Who is the most likely candidate for the claim? The mufti of Jerusalem is said to be "Hitler's Arab friend", but it has to be someone from Iraq the most likely person seems to be Rashid Ali al-Gaylani. Unfortunately, neither he nor his entourage where in Berlin to advise for any speeches:

On 25 August 1941, British and Soviet forces invaded Iran and removed Reza Shah from power. Gaylani then fled to German-occupied Europe.

Conclusion

Drawing together his remnant Catholic educational background, the lack of evidence for the quote as claimed, and the non-sensical construction to relate a surah, this surah – or in fact anything from the significant part of it relating to the moon – with events that pertain to the invasion of the Soviet-Union, I conclude that this "quote" is not a misunderstanding, nor a mistranslation. It is an invention.