Both are correct, but "in" sounds more natural.
Incidentally, if you were in the UK, you would say, "Put it in the fridge". :-)
english.stackexchange.com/users/303/Steve Melnikoff
I agree with Steve Melnikoff that put something in sounds more natural.
Put something into puts the emphasis on the motion, put something in on the target.
Examples for use of put something into that portray that fact are: put something into motion (formerly motionless) put into play put into effect
There is a verb put in something , meaning to apply, plant, spend (time/money), introduce, (nautically) enter a harbor or port.
english.stackexchange.com/users/None/
For me, "into" implies motion (or perhaps motion in a metaphorical domain).
If you are using a verb which already has motion (such as "go" or "put")
it is not necessary, and I think is slightly more formal, than "in".
There is rarely a difference in meaning, but where the "in" could be paraphrased as "from" or "out of", then "into" is definitely wrong.
So "I got the toy in the box" and "I got the toy into the box" are both fine, but "I found the toy into the box" doesn't make sense, and "I read the story into the book
" is definitely wrong (unless you mean somehow the technical computer sense of read, where you can "read" something into somewhere, as well as out of somewhere).
english.stackexchange.com/users/547/Colin Fine
I think that we use into when we want to emphasize the fact that an object or a person changes their place.
For example, we would say He got into the train instead of He got in the train but He is in the train instead of
He is into the train
But again, this is something I'm trying to remember properly, and I'm not absolutely sure.
english.stackexchange.com/users/2310/Grewe Kokkor
Both usages are fine as per Wikitionary, based on the implied meaning: in: into Less water gets in your boots this way.
into: going inside (of)
Mary danced into the house.
english.stackexchange.com/users/None/
Possible Duplicate:  When should “into” be used rather than “in to,” and vice versa?     I am sorry if the question is silly, but I think I heard both options spoken by native speaker. So, which one is correct?
"Put it into the refrigerator" or "Put it in the refrigerator"?
english.stackexchange.com
<word-choice><grammaticality><prepositions>
english.stackexchange.com/questions/5653
english.stackexchange.com/users/2270/brilliant
There is rarely a difference in meaning, but there are some circumstances where "into" is definitely incorrect. However, in this context, some people would claim that "in" sounds more natural. "Into" potentially puts more emphasis on the motion of the object in question. There is a difference of opinion on whether to use "in" or "into".  "Into" for some seems to put extra emphasis on the motion, particularly using a verb like "go" or "put". Wicktionary uses into regarding movement towards.
There is rarely a difference in meaning, but there are some circumstances where "into" is definitely incorrect. However, in this context, some people would claim that "in" sounds more natural. "Into" potentially puts more emphasis on the motion of the object in question. Both are correct, but "in" sounds more natural.  Put something into  puts the emphasis on the motion,  put something in  on the target.  
Both are correct, but "in" sounds more natural.  To put something into  puts the emphasis on the motion. To  put something in  emphasizes the target.   There is rarely a difference in meaning, but there are some circumstances where "into" is definitely incorrect. However, in this context, some people would claim that "in" sounds more natural. "Into" potentially puts more emphasis on the motion of the object in question.
Both are correct but "in" sounds more natural. The use of "into" adds extra emphasis, suggeting motion towards. Both usages are fine.
Both are correct, but "in" sounds more natural.   Put something into  puts the emphasis on the motion,  put something in  on the target.  
Some people would claim that, in this context, "in" sounds more natural. "Into" potentially puts more emphasis on the motion of the object in question. There is rarely a difference in meaning but there are circumstances where "into" is definitely incorrect.