Are you trying to set the connection string dynamically?

Yes. I'm using umbraco.GlobalSettings.DbDSN.

But when site loads first time and umbracoDbDSN in the web.config and a connection string which is set dynamically are not the same I get exception. Because actually Umbraco uses the old connection string. Second site loading is successful.

I don't know the answer to this but would suggest it's probably worth looking at the source code.  The installer wizard must be doing this so I would expect you could see what's going on there and utilise it for your own requirements.  In case you are unaware the code is available here.

Have a look at this Stack Overflow question. It seems to have the answer you are looking for:

Configuration config = ConfigurationManager.OpenExeConfiguration(ConfigurationUserLevel.None);
config.AppSettings.Settings["umbracoDbDSN"].Value = “New Connection String”;
config.Save(ConfigurationSaveMode.Modified, true);
ConfigurationManager.RefreshSection("appSettings");

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