There are countless of push-up variations one can do, each with its own benefits.
One of those is the “Gymnastics Push-Up”.
Carl Paoli, from gymnasticswod.com has three very good videos explaining in detail how to do them correctly.
I really like this variation, as it engages the core so much more, and if done right makes it virtually impossible to fall back into an anterior pelvic tilt.
Push Up Progression Pt.1
Push Up Progression Pt.2
Push Up Progression Pt.3
Its a pretty good read.
Main take away: set up in an arrow form and avoid anterior pelvic tilt (APT).