Can an airliner fly out of a downburst?
First take a look http://www.erh.noaa.gov/cae/svrwx/downburst.htm for info on downbursts.

An aircraft encountering a microburst (the more localized form of a downburst and that which airline crews train for) can counter it in two basic ways:


establish, if possible, a climb rate through the descending air that is greater than the downward vertical speed of the air.
fly through the column of descending air to air that is not descending or is at least descending at a lesser rate


And, of course, do this without stalling the airplane, especially as you come out of the burst. A practical scenario is that your max climb rate through the descending air can at least reduce your descent rate toward the ground enough that you will be able to transit the burst before hitting the ground.

I was never in a full blown microburst for real, but I practiced recovering from them many times in simulators. For 747-100/200 aircraft in the 1990s the procedure was:


thrust levers all the way forward
raise the nose until you get the stick shaker
fly the aircraft straight ahead while keeping the speed solidly in the stick shaker range.
do not change the aircraft configuration, in other words don't mess with the flaps or the gear.


In simulators it's a wild ride. You can tell how far into the stick shaker range you are by the frequency of the shaker by feel and by its noise. The closer you get to an actual stall, the faster the stick shaker will go, so the idea was to keep the shaker on but not too fast. Since the airspeed will be varying rapidly, you have to do that with large pitch changes.

There was controversy back in the 1990s as to whether what we were trained to do was actually the best way, especially as to not changing the aircraft configuration and how deep to go into the stick shaker. Perhaps now the thinking is different.