Looks like it is possible. Take a look at "Microsoft Knowledge Base Article - Q252987" on support.microsoft.com. In case if you won't be able to find this article, here is the code:
<%@Language=VBScript %>
<% Option Explicit %>
<html>
<head>
<title>Q252987 Sample Code</title>
</head>
<body>
<%
Const DTSSQLStgFlag_Default = 0
Const DTSStepExecResult_Failure = 1
Dim oPkg, oStep, sMessage, bStatus
Set oPkg = Server.CreateObject("DTS.Package"

oPkg.LoadFromSQLServer "MyServer","MyUser","MyPassword",DTSSQLStgFlag_Default,"PackagePassword","","","MyPackage"
oPkg.Execute()
bStatus = True
For Each oStep In oPkg.Steps
sMessage = sMessage & "<p> Step [" & oStep.Name & "] "
If oStep.ExecutionResult = DTSStepExecResult_Failure Then
sMessage = sMessage & " failed<br>"
bStatus = False
Else
sMessage = sMessage & " succeeded<br>"
End If
sMessage = sMessage & "Task """ & oPkg.Tasks(oStep.TaskName).Description & """</p>"
Next
If bStatus Then
sMessage = sMessage & "<p>Package [" & oPkg.Name & "] succeeded</p>"
Else
sMessage = sMessage & "<p>Package [" & oPkg.Name & "] failed</p>"
End If
Response.Write sMessage
Response.Write "<p>Done</p>"
%>
</body>
</html>