Amazon.com Inc. tested a pop-up feature on its app that in some instances pitched its private-label goods on rivals’ product pages, an experiment that shows the e-commerce giant’s aggressiveness in hawking lower-priced products including its own house brands.

The recent experiment, conducted in Amazon’s mobile app, went a step further than the display ads that commonly appear within search results and product pages. This test pushed pop-up windows that took over much of a product page, forcing customers to either click through...