Learn English – One word for the type of smile which suggests “I’ve got you! Now you can’t escape!”

Escapism comes close.

an activity or form of entertainment that allows people to forget about the real problems of life

_{Source: http://www.merriam-webster.com/}

From Wikipedia:

Many activities that are normal parts of a healthy existence (e.g., eating, sleeping, exercise, sexual activity) can also become avenues of escapism when taken to extremes or out of proper context.

Also, avenue of escape is used colloquially.

Alcohol and other drugs provide an avenue of escape -- a way to elude the mundane -- the boredom and routine of everyday living, as well as a means to temporarily evade difficult issues and problems.

_{^{Source: "Counseling Addicts and Offenders: A Guide to Criminal Justice Counseling" edited by Paula M. Potter}}

You can even use escape itself.

A means of obtaining temporary freedom from worry, care, or unpleasantness: Television is my escape from worry.

_{Source: http://www.thefreedictionary.com}

Additionally, I'm going to talk about a rare word which is used figuratively: nepenthe. As it is not a common word, you wouldn't hear it in everyday speech. The word first appears in the fourth book of Homer's Odyssey.

Wikipedia defines as:

... a medicine for sorrow, literally an anti-depressant – a "drug of forgetfulness" mentioned in ancient Greek literature and Greek mythology, depicted as originating in Egypt.

Figuratively, it means "that which chases away sorrow"; νη, ne, i.e. "not" (privative prefix), and πενθές, from πένθος, penthos, i.e. "grief, sorrow, or mourning"; so, literally, it means 'not-sorrow' or 'anti-sorrow'.

An example from "The Gilded Tongue: Overly Eloquent Words for Everyday Things" By Rod L. Evans:

For many people, alcohol is a nepenthe, used temporarily to escape their problems.

How about sag? Your mouth can sag and your whole face and body, in fact, can sag, when you feel sad.

When your mouth sags, it goes down.

You can also say that your spirit sagged.

I think this is related to being depressed. If you think about a depression being a low spot.