This is a great answer to the question taken from the Karaite site 'Karaitekorner'
Beginning or End of the 14th?
The Torah commands that the Passover sacrifice be brought "In the First Month on the Fourteenth Day of the Month between the two evenings" (Lev 23,5). It is unclear from this verse whether what is being referred to is the period of dusk at the beginning of the 14th or the period of dusk at the end of the 14th. Lev 23:6 continues that the Feast of Unleavened Bread is "on the Fifteenth Day of this month". From this verse it appears that the Passover Sacrifice is to be brought at sunset at the end of the 14th and eaten on the night of the 15th. This is confirmed by Dt 16:4, which commands us regarding the Passover Sacrifice: "and there shall not remain of the meat that you slaughter at evening on the first day until the morning." We see that the entire Paschal lamb must be consumed on the following night it is slaughtered and none of it may be left over until the morning (see also Ex 12:10,22). For our purposes what is significant is that the verse describes the Passover sacrifice as being slaughtered "at evening on the first day".
The passage in Dt 16:1-8 is talking about the Feast of Unleavened Bread and there can be no doubt that "the first day" in v.4 refers to the first day of Unleavened Bread. We have already seen in Lev 23:6 that the First Day of Unleavened Bread falls out on the 15th of the First Month. When we look at Lev 23:5-6 and Dt 16:4 together it becomes clear that the Passover Sacrifice is brought at the end of the 14th of the First Month between the two evenings and eaten that same evening on the 15th of the First Month. The period of "between the two evenings" is reckoned as both the end of the 14th (Lev 23:5) and the beginning of the 15th (Dt 16:4)!
It is not unusual for the Torah to refer to "such and such a date at evening" and to mean the evening that ends that day. In Lev 23:27 we learn that the Day of Atonement occurs on the 10th day of the Seventh Month. A few verses later the Torah makes clear what is meant by the 10th day: "and you shall afflict your souls on the ninth of the month at evening, from evening to evening you shall observe your Sabbath" (Lev 23:32). So we see that to fast on the 10th day means to fast from sunset on the 9th until the following sunset. In this verse "the ninth at evening" refers to the onset of evening at the end of the 9th, not the beginning! So the fast of the Day of Atonement on the 10th of the month runs from sunset ending the 9th until sunset ending the 10th (see also Ex 12:18). Similarly, the 14th between the two evenings in v.5 of the same chapter refers to the end of the 14th, not the beginning, as confirmed by Dt 16:4.
"and there shall not remain of the meat that you slaughter at evening on the first day until morning" Dt 16:4