According to fairmormon.org:
Latter-day Saints do not base their worship practices on an analysis
of early Christian history, or on the comments of scholars in Biblical
commentaries, though these sources can confirm Church teachings.
Rather, the Saints follow the guidance of a living prophet. However,
it seems clear that the Latter-day Saint practice of observing the day
of rest and worship on Sunday—like most of the Christian world—is
consistent with the earliest Christian practice of which we have
record.
See Brian's answer
Interestingly, however, the most important aspect of Sabbath worship
for the LDS seems to be the worship, and not the day on which it is
held. Most LDS worship occurs on Sunday. General Authorities, who must
often travel on conference assignments on Sunday, fast and receive the
sacrament weekly on Thursdays. Church branches in Israel worship on
Saturday. Branches in Muslim countries, such as Egypt, meet on Friday,
the Muslim holy day. Wrote one account of the Church in Israel:
Jerusalem is home to three major religions: Judaism, Christianity, and Islam. None of the three shares the same day of worship. Islam
recognizes Friday as a holy day, Judaism celebrates the Sabbath on
Saturday, while Christianity generally adheres to a Sunday day of
worship. These differences posed significant challenges in the lives
of the Saints living in the Holy Land, and David Galbraith posed
questions regarding this matter to President Lee during the Prophet's
visit to Jerusalem [in September 1972].
Following President Lee's visit, branch president David Galbraith wrote a letter to the First Presidency wherein he outlined four major
concerns and formally recommended that the day of worship for
Latter-day Saints in the Holy Land be changed. The four concerns were
as follows: First, for the Jews, public transportation ceases on
Saturday, stores and places of entertainment are closed, and in
Jerusalem the streets are full of families going to and from their
synagogues. Second, Sunday, on the other hand, is a normal working
day. Those attending the universities have classes, many of the
children have school, and, in fact, everyone except those in the
diplomatic corps have other obligations on that day. Third, the
members were scattered throughout the country, and the majority relied
on public transportation. It would be impossible to hold late
afternoon or evening services on Sunday. Fourth, the members of the
Church had been holding their meetings on the Jewish Sabbath rather
than Sunday for some time with at least the tacit approval of the
mission.
Two months after President Harold B. Lee's visit to the Holy Land, he authorized President David Galbraith to conduct worship services in
Israel on the Jewish Sabbath (Saturday). This authorization is dated
November 20, 1972. This decision in Israel served as a precedent to
include Friday observance as a day of worship in countries of
primarily Islamic populations, such as Egypt and Jordan.
Note that this is not an official source, but I have no real reason to doubt that this event happened as reported.
To finish, it seems like there is no clear commandment in Doctrine and Covenants to hold the sunday as Sabbath, but we do hold the sunday as Sabbath because the prophets have taught us so - and it seems to be consistent with the NT, or in other words, that has been the prophets' understanding of the scriptures so far (see Brian's answer).
