I found the solution.
In the Utilities class generated by the Visual Studio Template there is the following static method:
public static DTE2 GetDTE(object dataContext)
{
ICustomTypeDescriptor typeDescriptor = dataContext as ICustomTypeDescriptor;
Debug.Assert(typeDescriptor != null, "Could not get ICustomTypeDescriptor from dataContext. Was the Start Page tool window DataContext overwritten?");
PropertyDescriptorCollection propertyCollection = typeDescriptor.GetProperties();
return propertyCollection.Find("DTE", false).GetValue(dataContext) as DTE2;
}
By passing in the DataContext from my Control into the GetDTE() method I can do this:
var dte = Utilities.GetDTE(dataContext);
dte.Solution.Open(fullPathToSolution);