This one is harder than using only the hose as the Extinguisher runs out much quicker. What I did was on level 1-1, I went in to a room and used only L2 to put out all the fires. Then I went back to the initial room to refresh my extinguisher contents before moving rooms. Takes longer, but means you don't risk running out and the fire spreading to a room you have already cleared whilst you trek back to get more supply.
Finishing off the level will unlock this providing you only used L2 for putting out the fires.
I would advise doing this on a dedicated run on level 1-1. Ignore workers and the cat unless you can easily grab them. 5 minutes should be enough to clear the first level. You will end up low on time for the next level with this tactic, hence the reasoning for a dedicated run.