There are a couple of facets to the answer.
First of all, the Gospels themselves attest that the Resurrection occurred (or, at any rate, was discovered) on the “first day of the week”:
Now after the Sabbath, toward the dawn of the first day of the week, Mary Magdalene and the other Mary went to see the tomb (Mt 28:1).
When the Sabbath was past, Mary Magdalene, Mary the mother of James, and Salome bought spices, so that they might go and anoint him (Mk 16:1).
But on the first day of the week, at early dawn, they went to the tomb, taking the spices they had prepared. (Lk 24:1).
The Sabbath is our Saturday, and it was considered the last, or seventh, day of the week. (See Gen. 2:2.) Therefore, the “first day of the week,” when the Resurrection is reported to have taken place, is the next day, our Sunday.
Secondly, the Jews, then as now, reckoned their days from sundown to sundown, rather than from midnight to midnight (as the O.P. points out). (For this, see Lev. 23:32, as well as “The Time of the Crucifixion and the Resurrection”.)
Finally, when computing time, the Israelites generally would count any part of a day as a full day, even if that part was very small. (Among many examples of this way of reckoning, the Jewish way of saying “a week later” was to say “in eight days,” as, for example with regard to circumcision “on the eighth day.” See Gen. 17:12; Lev. 12:3; Lk. 1:59 and 2:21.)
Hence, Jesus was crucified on the Parasceve (the Day of Preparation, or Friday) before sundown (see, e.g. John 19:14); he lay in the tomb through the entire Sabbath; and he rose early Sunday morning, as we saw above. That, in the Jewish reckoning, works out to three days.
In other words, where the O.P. lost count is that “day one” would actually be the few hours between Jesus’ death and burial and the beginning of the Sabbath on Friday evening.