Use of punctuation marks is determined by style sheets. With regard to punctuation, spell-checkers are a form of style sheet that you are free to ignore. Because I find it natural to insert a pause after "said" when reading the sentence as written aloud, I'd normally follow the cadence of spoken English by inserting a comma, just as you have done. (Of course, I would not do so if writing for a publication with a required style sheet that prohibited a comma here.)
With that said, what your spell-checker has done is to alert you that the sentence as written is confusing. The word order seems to imply that she said to come in before she opened the door. If that is the odd situation being described, then it would be much clearer to say "'Come in!' she said before even opening the door." If the more normal situation obtained where the invitation to enter occurred after the door was open, then it would be clearer to say "'Come in!' she said after opening the door."
When a spell-checker makes a suggestion, you may not like the suggestion, but you may decide that some other alteration is advisable.