Referee wants me to do more experiments, but I already submitted these results to another journal I submitted two articles to two journals as corresponding author in short succession. Now, I got back the referees' comments to the first one and I am facing a problem: A referee requested us to do more experiments and claimed that the article lacked experimental evidence in a certain aspect.

Actually, these experiments were already carried out, but were described in the other article that is more focussed on the experimental part of our studies (while the first one is more of a general proof of concept). 

Obviously, I am facing a dilemma now: I don't think I can use the experimental results the referee requested in my first article, because I would be committing self-plagiarism. On the other hand, if I do not use those results, I fear the referee might reject the article.

A colleague suggested to tell the referee exactly that and show him/her the pre-print of the second article. But that does not really change the fact that my article seems to lack experimental evidence?

I could argue that these requested experiments/results are not essential to the submission. But I already kind of evaded the other big remark of the referee, thus almost changing nothing in my article. I feel this might be insulting to the referee, especially because I think he/she did a good job and really took time to review the article.

What do you guys think is the right course of action here? How do I argue my case best so that the referee is pleased? Could I cite the pre-print of my second article in the first article? Or can I reference in the article that there are experiments to be released in another submission that will handle the topic the referee spoke about?
I think the normal thing to do is to forward-reference the experimental results. 

That is, you can briefly outline the experimental results as per the current referee's desire for more information, and reference it as "paper submitted to xyz journal, 2/9/2017."  Down the road, if the experimental paper turns out to be accepted before the theory paper, all the better and you can actually reference it as such.

As a caveat, it may also be true that either paper is too thin for separate publication, and the two should be combined into one. This sounds like what the referee is getting at. From your point of view, having one strong paper may be more desirable than two weak ones. 

Finally, most often I have seen two papers of this type published back-to-back in the same journal.  But even if this is appropriate here it sounds like it may be a bit late to go this route, unless you were to withdraw one and resubmit the pair. (BTW, If you do this, an explanation to all the editors involved should definitely be included in your correspondence.)