Technology is applied science.
Science is the application of the scientific method to understand something.
Like most cultures through human history, I think you can merely show that, over time, the technologists will simply understand the "magic" using scientific principles. As aspects of the magic are known and understood, incorporating them into society and technology will be little more than some graduate student writing a paper covering the research they did that shows a previously considered "magic" is simply a physical process of the universe they live in.
Fire was magic at one point, and difficult to understand and control. But now we know it to be the rapid oxidation (ie, combustion) of flammable materials, and we not only know how to start it and stop it, but we can direct it and it is the primary choice for safe, inexpensive heat worldwide.
Now, they are fighting a war, and expediency might require some prodding of the graduate students to make a few leaps of the imagination so the technologists can use magic without appearing to subscribe to the notion of magic. At its base, though, magic exists and they merely need to show that an effect is repeatable in testing, and how to create the effect. It may not even be necessary to show or understand the mechanism - as long as others can verify through similar testing that an effect is reliable following a sequence of steps or using a set of materials, then it is scientifically usable without worrying about a possible connection to a religious belief. Showing that anyone can perform the effect with moderate instruction will go a long way to allaying the fears of the people with regard to "magic".
This won't necessarily work quickly if the magic requires an extreme level of skill/expertise/training. If there are few mages on the opposite side, and magic is naturally restricted only to those who have dedicated themselves to constant training and learning, then it will be much harder for said graduate student to conduct the research. However, it can still be done - or at least handwaved by said student - in order to win the war, while telling the people that the hard to understand papers turn magic into a scientific principle.