As I know, ''talk (one's) way into something/someplace'' means that a person uses charming or persuasive speech in order to achieve something or enter into someplace. In this example, it might just mean that these spies ''had ten minutes to use their persuasive abilities so that they could enter the stranger's flat...'' As Amy just said above, it's commonly used as the opposite of ''talk (one's) way out of something/someplace'', so, in this case, the sentence should also read: ''... they had ten minutes to manage to get themselves into the flat of a complete stranger...'' That's how I understand the usage of this expression. Hope I've been of some help to you!

Convince. It is usually used in the opposite "talk their way out of" as in "talk themselves out of trouble." Example : My parents saw the dent in the car, but I talked my way out of it by saying it must have gotten hit by a cart in the parking lot. In the way you are using it - Example : I tried to convince her to let me in the program by telling her all the ways it would benefit her to let me join. I can't believe I talked my way into such a prestigious club! (But again, it isn't used in this way often)