The winter solstice is indeed the day with the least amount of sunlight. However it is not the earliest sunset which is generally around Dec 7. From about the 7th to the Solstice, the amount of daylight lessens however sunset is later.
The chart shows Las Vegas sunsets and day lengths.
Earliest sunset 4:26 Dec 1-10.
Least amount of daylight, Dec 21. Sunset is 4:30.
This pattern is true in northern hemisphere although times differ based on place.