Yes, a dishwasher almost always uses less water than manual washing.
According to a study published by the University of Bonn, both the energy and water consumption of a dishwasher is better:
As regards the normal household practice of
washing small amounts of dishes and heavily
soiled articles, our comparison confirms the
advantages of automatic over manual dishwashing when comparing the average behaviour in
manual washing with a fully loaded average
dishwasher. These advantages can be identified
as lower consumption of water and energy and especially as
better cleaning results and significantly lower amounts of
manual working time needed.
As they say, the reason is mostly that smaller amounts are washed when handwashing. The average water consumption for 12 place settings is on average 83-121 litres (depending on whether all the plates are washed together or in 2 portions) when washed by hand and 20 litres when machine washed. They do however say that there's a huge spread on how much water is used when washed manually:
The water and energy measurements (Fig. 5 and 6) again
show a very wide distribution of consumption values, ranging from four to 90 l and from 0.03 kWh up
to 2.6 kWh for washing a pair of place settings.
As this is for only 2 place settings, even the most efficient manual washers are unlikely to achieve the same efficiency in cleaning as a dishwasher.