I'm an undergraduate psychology student currently finishing up my dissertation, but a few days ago, while going through my SPSS output, I realized my Levene's test is significant, but my data is normally distributed. Is there any way I can justify that it is ok to still perform a two-way between-subjects ANOVA?
As pointed out correctly by Erik, the question is how bad the deviations from the assumptions actually are. In other words, how big is the difference between the smallest and largest variance. In your case it sounds as if they are not that big. Then it is no problem. However, you have to take differences in the samples size into account. If I remember numbers for these issues can be found in the book on ANOVA by Tabachnik & Fidell.
Additionally, as I already said here, using the Levene test (or actually any null-hypothesis significance test on the assumptions) heightens your overall probability to commit an alpha error and is therefore not recommended. The other answers in the linked thread could also be interesting to you.