Insta is right, a 555 is simple to setup for this and costs less than a dollar. I was thinking you were including something like the 555 in your considerations, but just in case here is a possible transistor only solution (parts shown and values are guidlelines, circuit is rough and could be improved, but it's gives the idea - time will vary with components chosen, so you may need to tweak things a bit, add a pot instead of the R1/R7, etc)
Circuit:
Simulation:
L1 is your relay coil, and is powered for roughly 60 seconds on the 12V supply dropping out.