According to those who pray Mincha only up to Shekiya (sunset), are you required to finish Mincha before Shekiya or just required to begin it before Shekiya? (Sources.)
The simple answer (which is used for so many other things as well) is from Bilaam. No wonder you ask it now this week.
The gemoro says in B'rochos 7a that Bilaam could work out the exact microsecond when G-d was angry. Tosfos asks what Bilaam could possibly say in that microsecond. One answer given is that he could say just one word: Destroy. Another answer given is that, once he started speaking his curse at the exact microsecond, he could continue speaking (and cursing) as long as he wanted.
So according to Tosfos's first answer you have to finish in time, but according to the second you don't.
The same as when G-d is angry, if you start 'on time' that means catch the second you can keep his anger 'going'. How much more so that when he is not angry and awaits our t'fillos to fulfill them that if one starts on time, hashem will have patience to wait until we finish.
Aruch HaShulchan 110:5 says that if one began on time, even if he finished after the Zeman it is considered Tefila B'Zmana.
However I have heard that the Mishna Berura disagrees and says that one must finish the entire Tefila by the Zeman. (will put in source as soon as it is found)