There are many ways to politely and humbly respond to a compliment. For example,
In this expression, the first sentence shows gratitude for the complement, and the second sentence means that (the speaker successfully accomplished the task or something) because there was Mr. Miyazaki’s help or support. This expression is polite and humble. The speaker implies that Mr. Miyazaki deserves the complement, and thanks both her boss and Mr. Miyazaki.
If your boss is the one who helped or supported you, say the second sentence first, like so.
○○ indicates your boss’s name. If your boss’s position is not 部長, replace it with an appropriate job title. Or if you usually call your boss ◯◯さん, just call your boss the same way as usual.
This expression implies that the 部長’s support brought the success and the speaker is thankful for it.
Another way of saying the same thing as (A) is
Although this (B) expression has the same meaning as (A), おかげさまです can imply the help or support of not only the person the speaker is talking to, but also other people. So, おかげさまです can mean that (the speaker could do an excellent job or something) because there were the people (including the listener) who helped or supported the speaker.
「おかげさまです」 can be used for expressing something occurring in the present as well. The past form of です is でした. However,「おかげさまでした」 is not common.
Another expression having the same nuance as (B) is
Please be careful in using さま. 「○○さんのおかげさまです。」 does not sound natural in many situations. ○○さんのおかげです or おかげさまです is more common and safer to use.
It’s more formal if おかげさま is said with the excellent outcome which the support brought. For example,
This おかげさま also means the listener’s support, or the listener’s and other people’s supports. It depends on the situation.
Nice and polite people use these おかげ sentences above even in the situation where nobody helped or supported them actually. Because it’s polite thing to say, and a way to keep a good relationship with other people.
By the way, when saying ありがとうございます or a おかげ sentence, Japanese people generally do Eshaku(会釈) which is a kind of body language. Eshaku is a shallow bow used in daily life to express various feelings. You work with Japanese people so I suppose you already know Eshaku, but if you like to see what Eshaku is like, this video may be helpful. The 15-degree bow is Eshaku.
Eshaku or a deeper bow can express that you are truly grateful. So, saying those sentences with Eshaku is good and natural. There would be no contradiction between your verbal and non-verbal expressions if you say it and do Eshaku.
If you’re interested in other ways to respond to a compliment about what you did at work, the following expressions are also polite, humble and good to use.
These expressions imply that the speaker thinks it was not perfectly good so s/he keeps learning and trying to do her/his best.
If the complement is really too much, I recommend using a formal word 恐縮, which is a convenient word to express that the speaker is very afraid (1) because something, which someone gave the speaker, seems too good to him in his mind, or (2) because what the speaker did or is doing is not appropriate and did or may bother someone, (3) because the speaker needs to ask someone to do something which the person is probably not interested in, (4) etc. (1)-type 恐縮 can fit your situation.
恐縮 is used to be humble, and generally said with the reason why the speaker is being very afraid. Saying the reason is a part of politeness in many cases.
For example, if your boss compliments you on your success, you can also answer like so.
This expresses that the speaker is thankful for the compliment, but also afraid to accept it, because Mr. Miyazaki is the one who greatly contributed to the success. (The dot symbols …… in this sentence indicate the silence with sincere facial expression.)
If there are several 恐縮 reasons, it can be said something like this.
And, the following conversations are examples of answers to 「早いね」.
(and have a little talk to be polite.)
早いね can imply various things. So, the answers to it vary as well.