It's a bit easier than wise in some aspects, because some of it can be glossed over. If the general arrives at the battlefield, takes in his retreating men, and raps out, "Gentlemen, that position on the right must be retaken!" -- you can then award him victory without going into details about how crucial it was. Or the engineer can glance over an engine and ask if the screws are a bit loose over there.
But not always. Sometimes you actually have to master the stuff. This calls for rip offs, big time. Find a past battle noted for strategic brilliance and use it. Perhaps have your character talk about how this situation has happened in the past.
One particular point is when the character's making scientific discoveries. The prize has to go to the novel where an alien was born in a society of flat-earthers. He discovered that the world was round, that it was orbiting another body, and that it was in danger of being destroyed through falling into Roche's limit, so he declared they needed a space program to escape. (AAAAARRRRRRGGGGHHHH!) The important thing to research is what arguments were made against it, and what was needed to refute them. If you have a Galileo figure, have someone point out if we went on a grand circular tour every year, we would be nearer some stars at one solstice, and nearer to others at the other, yet the stars always look the same to us, or that we all know the difference between sitting still and moving, and yet by his argument we are always moving and should feel it and see its effects. You can have the Galileo character claim that the tides are in fact caused by the water sloshing about due to the planet's motion. You can even have the opponent point out that we do not obstruct our vision when we want a clear sight, so why is he claiming that obstructing his vision with two warping pieces of glass makes his vision better? Indeed, that one might be best, because the evidence for the other two was unavailable in Galileo's time; his physics experiments are the foundation of physics, but none of his claims about the heliocentric system are considered evidence nowadays. You can have your character demonstrate that he sees more clearly through the telescope though.
I still remember reading G. K. Chesterton's George Bernard Shaw and his comment on how Shaw rejected that experiment the purported to show that acquired traits were not inherited by chopping off mice tails for generations and showing that the great-to-nth generation grew tails just as long -- which meant, Shaw pointed out, nothing about acquired traits, since having an experimenter chop off your tail is not an acquired trait, any more than being in a railway accident is. He had a point.
A problem more both subtle and pervasive is making the character appear smart even when not doing obviously smart things. Which, come to think of it, applies to wise characters, too, who should show good judgment regularly. But a prime example is Carthoris in Edgar Rice Burroughs's Mars books. At one point, he shows off two brilliant mechanical inventions he made, which appear to be sui generis. At no other time does he give the least hint of being an inventor. He's not even interested in machinery, or noticing machinery where it appears. Which is like many other alleged geniuses, who take out their brilliance only when it's needed. They will notice things in their field of interest like any other character, and they will notice things and make connections faster. And most of them will use unusual words -- casually, spontaneously, easily, because all those words are in their vocabulary, and not flagged as unusual there. And above all else, correctly. Malapropping is a grave danger there.