The phrase "managed to open" is correct and usable in written English.
It is usually used in a context that indicates a successful accomplishment. For example, "I managed to open the door with my spare key."
A handful managed to open small businesses.
Though expelled in 1907, they managed to open other hospitals and today maintain several institutions.
He seemed to elude defenders and continuously managed to open up space that shouldn't be there.
With their earnings, some managed to open businesses; some became customers.
The Lecht in the Cairngorms managed to open three lifts last Sunday.
Leopold struggled to find the right key and, after some fumbling, managed to open the door.
Had someone in one of the surrounding towers managed to open a window?
Weak with laughter, I managed to open my programme and underline his name.
Finally, the man fetched an attendant, who managed to open the door.
I managed to open the door, I couldn't see who was there.
After a few minutes, a guard managed to open the jammed door, and Tataki hurried toward the parking lot.
