It's old-fashioned if not outright archaic usage, and would be unfamiliar to most native speakers:
by (or in) virtue of
Because or as a result of:
virtue, Oxford Dictionaries. Oxford University Press, n.d. Web. 12 April 2015.
The full quote is:
At three o’clock, implore My mercy, especially for sinners; and, if only for a brief moment, immerse yourself in My Passion, particularly in My abandonment at the moment of agony. This is the hour of great mercy… In this hour, I will refuse nothing to the soul that makes a request of Me in virtue of My Passion.
Diary: Divine Mercy in My Soul, Faustina Kowalska
To make sense of it you also need to know that 'passion' is being used in a special sense:
The suffering and death of Jesus
passion, Oxford Dictionaries. Oxford University Press, n.d. Web. 12 April 2015.
So Kowalska writes that Jesus said to her (speaking in rather archaic language) something like this:
"At three o'clock, if someone makes a request to me where they give my suffering and death as the reason that I should grant their request, I won't refuse them".
An alternative interpretation could be:
"At three o'clock, if someone makes a request to me, I won't refuse their request, because of my suffering and death".
Kowalska's native language was definitely Polish; I'm not clear whether Kowalska wrote her diary in English or Polish; even if she wrote it in English, I'm not clear whether she's writing that Jesus spoke to her in English, or whether she's writing that he spoke to her in Polish and she's translated it to English. So translation may be an issue here. Anyway, it's a very formal and old-fashioned tone.