<DataGrid DataGridCell.Selected="DataGrid_GotFocus" ItemsSource="{Binding Source={StaticResource itemView}}">
    <DataGrid.Columns>
        <DataGridTextColumn Header="Nom" Binding="{Binding Path=Name}"/>
        <DataGridTextColumn Header="Age" Binding="{Binding Path=Age}"/>
    </DataGrid.Columns>
</DataGrid>

private void DataGrid_GotFocus(object sender, RoutedEventArgs e)
    {
        // Lookup for the source to be DataGridCell
        if (e.OriginalSource.GetType() == typeof(DataGridCell))
        {
            // Starts the Edit on the row;
            DataGrid grd = (DataGrid)sender;
            grd.BeginEdit(e);
        }
    }

<DataGrid DataGridCell.GotFocus="DataGrid_CellGotFocus" />

private void DataGrid_CellGotFocus(object sender, RoutedEventArgs e)
{
    // Lookup for the source to be DataGridCell
    if (e.OriginalSource.GetType() == typeof(DataGridCell))
    {
        // Starts the Edit on the row;
        DataGrid grd = (DataGrid)sender;
        grd.BeginEdit(e);

        Control control = GetFirstChildByType<Control>(e.OriginalSource as DataGridCell);
        if (control != null)
        {
            control.Focus();
        }
    }
}

private T GetFirstChildByType<T>(DependencyObject prop) where T : DependencyObject
{
    for (int i = 0; i < VisualTreeHelper.GetChildrenCount(prop); i++)
    {
        DependencyObject child = VisualTreeHelper.GetChild((prop), i) as DependencyObject;
        if (child == null)
            continue;

        T castedProp = child as T;
        if (castedProp != null)
            return castedProp;

        castedProp = GetFirstChildByType<T>(child);

        if (castedProp != null)
            return castedProp;
    }
    return null;
}

<!-- SINGLE CLICK EDITING -->
<Style TargetType="{x:Type dg:DataGridCell}">
    <EventSetter Event="PreviewMouseLeftButtonDown" Handler="DataGridCell_PreviewMouseLeftButtonDown"></EventSetter>
</Style>

//
// SINGLE CLICK EDITING
//
private void DataGridCell_PreviewMouseLeftButtonDown(object sender, MouseButtonEventArgs e)
{
    DataGridCell cell = sender as DataGridCell;
    if (cell != null && !cell.IsEditing && !cell.IsReadOnly)
    {
        if (!cell.IsFocused)
        {
            cell.Focus();
        }
        DataGrid dataGrid = FindVisualParent<DataGrid>(cell);
        if (dataGrid != null)
        {
            if (dataGrid.SelectionUnit != DataGridSelectionUnit.FullRow)
            {
                if (!cell.IsSelected)
                    cell.IsSelected = true;
            }
            else
            {
                DataGridRow row = FindVisualParent<DataGridRow>(cell);
                if (row != null && !row.IsSelected)
                {
                    row.IsSelected = true;
                }
            }
        }
    }
}

static T FindVisualParent<T>(UIElement element) where T : UIElement
{
    UIElement parent = element;
    while (parent != null)
    {
        T correctlyTyped = parent as T;
        if (correctlyTyped != null)
        {
            return correctlyTyped;
        }

        parent = VisualTreeHelper.GetParent(parent) as UIElement;
    }

    return null;
}

<ResourceDictionary x:Class="YourNamespace.DataGridStyles"
            x:ClassModifier="public"
            xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
            xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
    <Style TargetType="DataGrid">
        <!-- Your DataGrid style definition goes here -->

        <!-- Cell style -->
        <Setter Property="CellStyle">
            <Setter.Value>
                <Style TargetType="DataGridCell">
                    <!-- Your DataGrid Cell style definition goes here -->
                    <!-- Single Click Editing -->
                    <EventSetter Event="PreviewMouseLeftButtonDown"
                             Handler="DataGridCell_PreviewMouseLeftButtonDown" />
                </Style>
            </Setter.Value>
        </Setter>
    </Style>
</ResourceDictionary>

using System.Windows.Controls;
using System.Windows;
using System.Windows.Input;

namespace YourNamespace
{
    partial class DataGridStyles : ResourceDictionary
    {

        public DataGridStyles()
        {
          InitializeComponent();
        }

     // The code from the myermian's answer goes here.
}

<DataGrid.Resources>
    <Style TargetType="{x:Type DataGridCell}" x:Key="SingleClickEditingCellStyle">
        <EventSetter Event="KeyDown" Handler="DataGridCell_KeyDown" />
    </Style>
</DataGrid.Resources>

Private _endEditing As Boolean = False

Private Sub DataGrid_GotFocus(ByVal sender As Object, ByVal e As RoutedEventArgs)
    If Me._endEditing Then
        Me._endEditing = False
        Return
    End If

    Dim cell = TryCast(e.OriginalSource, DataGridCell)

    If cell Is Nothing Then
        Return
    End If

    If cell.IsReadOnly Then
        Return
    End If

    DirectCast(sender, DataGrid).BeginEdit(e)
    .
    .
    .

Private Sub DataGridCell_KeyDown(ByVal sender As Object, ByVal e As KeyEventArgs)
    If e.Key = Key.Enter Then
        Me._endEditing = True
    End If
End Sub