Do White Bolts NOT Count as Being Detected

General Concept

The "trick" is not having them die and choosing to take more efficient paths to this end so as to also collect enough refugees and not run out of time. This involves protecting them from the constantly spawning enemies and the randomly placed meteors. Having sufficient crowd control through abilities and monions/followers is really the only way to keep the enemies at bay. Movement speed bonuses are useful if you need to run back and help out as well as get to the refugees faster, but the refugees don't go faster because of this. As far as the meteors are concerned, you just have to keep your eye out and back off if you don't think the refugees will make it past an oncoming meteor. As far as time, try to get reasonably-sized groups of refugees together in each trip in order to take fewer trips.

I've completed the requirements for this achievement several times with each class on single player. Witch Doctor is by far the easiest class to earn this achievement with. All of the followers work very well so take whichever you like best, but I prefer Scoundrel's speed and crowd control and Templar's aggro gain and crowd control. Crowd control abilities that stop/slow/distract enemies can be useful here as can abilities that increase movement speed. Any of the class' minion summoning spells are great for distracting the enemies, especially if the minions can be targeted by enemy attacks (monk and witch doctor).

The AI pathing for the refugees is pretty bad and they sometimes wander or run off to the middle of nowhere (or to try and fight the enemies?) and that's often your worst enemy. Learning when/where the enemies spawn is helpful as the worrisome spawns are always the same and triggered by your actions so you can learn where to attack. The meteors are your second biggest worry as they seem completely random.

My Strategy

I'm sure there are better strategies, but the approach that I took to consistently complete this achievement with each of the classes involved 4 trips and was as follows:

I added the unmarked refugees and drew out the paths in the above image.

1. While Asheara is still talking about the Iron Wolves saving people, get near the cart nearest to the sewer entrance (a bit north-west). If there's a meteor coming, let her talk, but once it hits, skip the speech and push the cart. You should be close enough to the sewer that he should very easily path to the escape. This is 1 refugee and should take like 5-10 seconds at most. At the very start, there are also 2 more refugees in the center of the beginning chaos in the middle of the right side of the bazaar and they tend to die very quickly from the enemies that spawn there and the frequent meteors, but if you are lucky and your minions, skills or the Iron Wolves protect them sufficiently, you can grab these 2 as well for a total of 3 refugees in about 15-20 seconds.
2. Run diagonally north-east through the chaos to the stalls and cart towards the top-center of the of the bazaar. Quickly push the cart and hang a bit to the north if there's a meteor so the citizen crawling out doesn't walk right into it, move to the left of the stall to make another crawl out and, including the fellow already standing there, you should have 3 with you at this point. Here's the first tricky bit. An enemy will spawn right in the middle of these stalls so attack as it spawns to kill it quickly. Run south through the chaos directly to the sewer, avoiding the meteors (If they mark the spot you are on, you should be OK to continue as long as nothing distracts the refugees. If you need, you could try hooking around the left where the first citizen was, but I've not had much luck that way, By this point you should have 4-6 and that last bit shouldn't have taken more than 20-30 seconds.
3. Run to the east, near the entrance to the bazaar to hit the cart just south-east of that archway. An enemy will spawn south of there while another will come up behind you - dispatch them. Next hit the cart north of the arch and continue north, dispatching the enemy that spawns there, to pick up the two along the way. Continue all the way to the north-east to grab the group of 3 up there. Head south, passing the north side of the fountain and another will crawl out. 1 or 2 enemies should spawn in your way and the meteors get annoying at this point. You will be guiding 8 refugees and the AI pathing can be uncooperative so this is the challenging run, following the same return path used in 2. You should have saved up to 11-14 by this point and this bit takes close to 1 minute.
4. Quickly, run west through the gate to the other end of the bazaar and a refugee may join you on the way (if not, he'll catch up on the return trip). Hit the cart to the north and then the one to the south. An enemy will spawn to your left - kill it. Continue south of the fountain to hit the north and then south carts there. Another enemy will spawn - it can die. Collect the group of 3 just north of here and hit the other cart for one more. Another will crawl out of a cart on the left At this point, you should have enough, but if you don't, there are 2 more in the stalls to the north. Dallying here can cost you though as it allows more enemies to spawn. When done here, return to the sewer for your reward. You can collect up to 12 more here and this should take 1 minute or so. The enemies in this run seem to spawn along your path rather than behind so they're easier to take out before they cause trouble.

There are a total of 26 refugees and you need to rescue 20 of them so you can let at most 6 die or not even grab them if you like. This should take about 2:40 at most of the 3:00 so you should have some leeway on the time. You could probably tag trip 2 on the end of trip 3 and save a trip, but with the enemy spawn frequency and locations as well as the argumentative AI, I found the refugees pathed better and died less when there were fewer of them for trip 3.

A post on Diablo III forums suggest a user got the achievement for killing Queen Aranaea, the spider boss in Act 1, so the other (sub) boses should get you the achievement as well.