This is a good question, I think, because it's important when thinking about international treaties to consider the enforcement mechanism described in the treaty. Here is the portion of the NATO treaty known as the collective defense article, Article Five:

The Parties agree that an armed attack against one or more of them in Europe or North America shall be considered an attack against them all and consequently they agree that, if such an armed attack occurs, each of them, in exercise of the right of individual or collective self-defence recognised by Article 51 of the Charter of the United Nations, will assist the Party or Parties so attacked by taking forthwith, individually and in concert with the other Parties, such action as it deems necessary, including the use of armed force, to restore and maintain the security of the North Atlantic area.

As you can see, the article does not necessitate armed assistance, but rather "such action [each nation] deems necessary, including the use of armed force." In practice, of course, it was expected that the United States would respond with immediate military action even if other nations could only provide some other form of support. Nevertheless, the treaty absolutely does not trigger a declaration of war.

Keep in mind, also, that even a direct attack on the United States by a foreign power does not automatically trigger a declaration of war. For example, when Japan attacked Pearl Harbor, the United States did not declare war on Japan until two days later, after Franklin Roosevelt formally called for the declaration and Congress nearly unanimously approved (a single Congresswoman, Jeannette Rankin, voted against the war, and she was voted out of office in 1942).

Such an event would be considered an emergency and the President of the United States of America would be allowed to command the armed forces of the United States. Interpretations of the separation of powers between the President's ability to command the armed forces versus the Congressional authority to declare war are complicated. The conflicts in Korea and Vietnam were never formally declared as wars by Congress -- the War Powers Act of 1973 grew out of this (especially Vietnam) and was set up as a check on the President's ability to use armed forces. It set up a required 48 hour notification of the outbreak and then a period of 90 days before Congressional war approval.

So, to answer your question, there would be little doubt that in the immediate moment of Soviet aggression, the President would have had full authority to command the armed forces, but, depending on whether or not the War Powers Act had been passed yet, some statutory requirements would be necessary.