A2A You do realize that you are asking an Harvard Fellow who did a post Doc at Harvard that worked with MIT research group this question. So my answer is very bias based on my experience in the 1980's. However, I will try to be objective in my answer.
Assuming you can actually get into either Harvard or MIT, then you would need to be more specific: Undergraduate or Graduate, Department, Major. Because "better" is applied to majors not to universities or institutions - MIT is an Institution. And at the graduate level, the term is meaningless because it is faculty who you do research with when the comparison are made, it is based on yours and theirs original research. Engineering is technical studies and a very board term that consists of a number of departments and faculty members.
For example, if you are talking about Computer Science, MIT based on research is ranked higher than Harvard, but not by much. Having sat in the Harvard classroom taught by an MIT professor, the lines become blurred at the research level. If you taking about Engineering Science where you are taking Science courses, Harvard is ranked higher than MIT. Harvard is Engineering and Applied Science, see Harvard John A. Paulson School of Engineering and Applied Sciences as compared to MIT Massachusetts Institute of Technology | Best Science School .... The issue is how theoretical are your engineering studies. Mine was Engineering Science, so Harvard was the better choice for me at the time.
Historically Harvard did not have an Engineering program until after WW II: It was Applied Science, see Harvard John A. Paulson School of Engineering and Applied Sciences Timeline. When I was there, Harvard had access to MIT researcher and vice versa, so what was important was the researchers you wanted to work with at the research level. There are no better researchers when you talk about Harvard or MIT: the one you choose to work with is the best for you, or had better be.
Bottom line: There is a thin line between Applied Sciences and Engineering. Depending on which side of that line will determine which is better. However, I must warm you that what distinguishes universities/institutions is the faculty, the students and the alumni. It is not uncommon for MIT graduates do an MBA at Harvard rather than Sloan. Because my specializations are in Computer Science, I would go to MIT; however, I went to Harvard because I wanted to work with a specific researcher.