The Sound Barrier is about engineers breaking things.
Tucker: The Man and His Dream
The General
Bridge Over the River Kwai
While I love The Rock, I’m not sure this fits the spirit of the OP, because Goodspeed has to become a traditional action hero to save the day, guns and hand-to-hand combat with VX gas included. His brains never really enter the equation - anyone could have disassembled those rockets and and destroyed the chips.
… both of which are about mathematicians, not engineers … 
Nearly every Star Trek: TNG episode is about how they engineer a solution to a problem. True, they just threw in gobbledygook names for stuff, but they always did it by presenting some sort of logic to the gobbledygook.
For example:
“We have a problem with invisible ships.” -> “We can set up ‘trip lines’ that detect the ships. If we arrange that into a lattice, we’ll be able to detect them.”
Definitely the engineer’s superhero! Makes his own superpowers!
The thing about Apollo 13 is that the engineers (and mathematicians, and technicians and “steely-eyed missile men”) solved the problem not by engaging in stereotypical “heroics” but by being very, very good at their jobs. I’d disqualify The Rock because, as already been mentioned, the heroics didn’t really involve engineering. I’ve never seen MacGyver, but I want to keep this to movies, so Star Trek might have to be rejected as well. And it’s a bit of a gray area, as the engineering is fictional, so it doesn’t have to make a lot of sense.
The Dish definitely fits the criteria. I wish Iron Man fit the criteria, but he’s an engineer in preparation for becoming a hero, but the heroics never directly involve his engineering skills.
I don’t know. The beginning of the first movie, when he put together the suit, was a lot of engineering and very little heroics. I liked that the suit was a bit of a kludge also - and enough of his background was given to make the work plausible.
Granted, but the heroic acts in Apollo 13 were engineers engineering, where Iron Man featured Tony Stark inventing something that allowed him to kick ass.
In Real Genius practically all the characters are engineers.
No Highway in the Sky starred Jimmy Stewart as an Aeronautical Engineer.
Absurdly enough, the eponymous character from the “Prince of Persia” movie could qualify. Mostly he kicks enemy ass by solving all sorts of structural puzzles.
The argument can be made that most of the ass-kicking was a result of the engineering stage. Put anyone in that suit, and he’d be equally heroic.
Jack Lemmon on The China Syndrome. Wilford Brimley, too. Nuke plant engineers save the world from a meltdown.
The British engineer-officers in Bridge on the River Kwai - perhaps not heroic, since their actions help the enemy, but they overcome their circumstances and restore some pride in their work.
The Aviator, Howard Hughes having been quite an accomplished self-taught engineer.
Any of the several Wright Brothers movies.