Title was full of promise, but we need more.

Ideally, we want to be able to determine what statements caused the log to grow so suddenly (and who issued the statements) and we want to do this automatically, not by running a query. i.e. at the time there is "excessive log file growth" alert the DBA providing a list of user-statements currently executing.

The tools and DMVs are out there but unfortunately do not provide a full picture (who, what, when, sql statements running). I think Profiler comes to the closest (Events "Log File Auto Grow" & "SQL:BatchCompleted") but even that involves monitoring and examining, not automatically alerting.