Since when do Moshe's hands make or break war?
Rashi (citing the Mishnah in Rosh Hashanah 29a): It was when Moshe raised his hands, that the people raised their eyes heavenwards and rendered their hearts subservient to Hashem. That was when they were victorious in battle; and when he dropped his hands, they fell. 1
Rashbam: "Moshe's hand" refers to the staff that he was holding; and it is human nature that when the army see their emblem raised, they fight harder and enter into a winning streak.
Targum Yonasan: When Moshe raised his hands in prayer, Yisrael were winning, and when he lowered them and stopped Davening, then Amalek were winning.
Maharal (Ohr Chadash p. 68): Tefilah is the only way to defeat Amalek. When "the voice is the voice of Yaakov" - in prayer to Hashem, then "the hands of Esav" have no power. 2 That is how Yisrael overpowered Amalek - though the Moshe's Tefilah. 3
See Torah Temimah, note 8.
See our comments to Bereishis 27:22.
Maharal (ibid. p. 54): The miracle of Purim is the foremost example of how when a calamity befell Yisrael, they cried out to Hashem, and Hashem heard their prayers. See Maharal as to what was unique about this Tefilah.
Rashi, citing Rosh Hashanah (29a), asked why the war depended on Moshe's hands. Is it hard to understand that Moshe's Tefilos helped to overpower Amalek?
Lev Eliyahu (p. 96): This war was not based on physical strength, but on Ruchniyus. Amalek, sensing that Yisrael were lax in Torah-study, traveled four hundred Parsah to Refidim - via witchcraft. Therefore, Yisrael needed to consecrate their hearts to Hashem. 1
Sichon and Og were stronger than Amalek; but they were physical wars, and Moshe alone was able to defeat them. The Rashba (in Chidushei Agados, Berachos 54b) queries this from the fact that Og had great merit, and also the merits of Yisrael and the Avos were needed to defeat him. Refer to Bamidbar 21:34:2:1* ; also see Bereishis 14:13:2.1:2* .