I have an SSIS data flow task with an OLE DB Destination component that inserts records into a table with a trigger. When I execute a normal INSERT statement a
INSERT
You can do this without using SQL Server Data Tool for Visual Studio by editing the dtsx File with Notepad (or any other Text editors).
Search for the following property:
TABLOCK,CHECK_CONSTRAINTS
and add the value FIRE_TRIGGERS as Diego already described.