make a run for it
to suddenly run fast in order to escape from somewhere or get to somewhere
When the guard turned away, the two prisoners made a run for it.
Let's make a run for it as soon as the rain lets up a bit.
Cambridge Idioms Dictionary, 2nd ed. Copyright © Cambridge University Press 2006
http://dictionaries.cambridge.org/define.asp?...

Also, make a run for. Run toward something. For example, As soon as it ended, they made a break for the door, or I'll have to make a run for the plane. The noun break here means “escape,” and both terms may be put as make a break or run for it, meaning “to escape or get away quickly.” For example, With the guards asleep, he decided to make a break for it, or The rain's stopped; let's make a run for it. [c. 1840]
The American Heritage® Dictionary of Idioms by Christine Ammer. Copyright © 2003, 1997
http://idioms.yourdictionary.com/make-a-break-for

Inside, “jazzeman04,” decides to cut our conversation short. He makes a run for it.
The detectives spring in to action arresting the man sitting in his car across the street, then “jazzman04,” and then the man parked on the side street. This all happens less than six minutes from the time the first car pulls up.
http://www.msnbc.msn.com/id/11152602/ns/datel...
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tür|men (salopp): sich aus einer unangenehmen Situation durch eilige Flucht befreien:
die Jungen türmten, als der Besitzer kam; aus dem Knast türmen; über die Grenze türmen
flüch|ten
a) [plötzlich und sehr eilig] fliehen; sich einer drohenden Gefahr durch Flucht (1) zu entziehen versuchen: 
vor den Soldaten, dem Hochwasser flüchten; über die Grenze ins Ausland flüchten; das Kind flüchtete ängstlich zur Mutter;
Duden - Deutsches Universalwörterbuch, 5. Auflage Mannheim 2003

türmen
fliehen ⇓
bei Nacht und Nebel türmen
nach N türmen
Na, Wachtmeister, nehmen Sie ihm die Kette ab, er wird schon nicht türmen — Fallada Blechnapf 470
• sich eilig entfernen, ausreißen ⇓
[sie] prügelten uns aus dem Lokal raus ... Wir türmten — Welk Grambauer 116
DWDS-Wörterbuch

"to make a run for it"
This phrase isn't always used in the sense of "to escape", but it often is. Maybe for the other cases something like "lossausen" would work? If someone else has a better idea, please suggest it.

At any rate, it's a fairly common colloquial expression in English and should be included in the dictionary. When I type "make + run" in the search box the only phrase that comes up is "to make a beer run". Hmm. Priorities? Sex, drugs, and rock-'n-roll are well represented on LEO, but other areas are strangely lacking. ;-)