Another 2 cents worth of advice after spending almost 2 days trying to figure out a solution:

My scenario was quite hectic since I had to to a migration that involved several environment changes:

Windows Server 2003 -> Windows Server 2008

SQL Server 2000 -> SQL Server 2005

32 bit -> 64 bit

I had to move a web app written in classic ASP to a 64 bit box and the app was executing DTS packages from inside a page. The error was "ActiveX Component can't create object: DTS.Package". I finally found about the limited/complicated support for DTS on 64 bit systems and I figured out that since there is no 64 bit version of the DTS dlls, clearly the DTS.Package was not to be found registered in COM where it was being looked for.

I had all the DTS 2000 components and backwards compatibility components installed, I was able to edit and run the packages under Mgmt Studio just fine, it was only the programmatic access that didn't work. The solution was pretty straightforward then: convince the IIS worker process to run in 32 bit mode so the DTS.Package was searched in the correct place in registry. Now I can only tell about IIS7 but I guess for older IIS versions solutions can be easily found. First, locate the application pool under which the web app is running, go to advanced settings and then make sure the "Enable 32-bit applications" setting is on "True". This way the w3wp process will run in 32 bit mode (easily verifiable in Task Manager where you will find a "w3wp.exe *32" entry) and the vbscript in the ASP page will look in the 32bit area of the registry for the DTS.Package COM object.

Hope this helps!