In his lost notebook, Ramanujan presented unique categories of remarkable infinite series, known as the Ramanujan-type Eisenstein series. The objective of this paper is to generate various differential identities related to classical $\eta$-functions and $h$-functions with the help of the Ramanujan-type Eisenstein series. Moreover, we present particular identities that incorporate Eisenstein series of different levels and $h$-functions, thereby establishing links between class one infinite series and $h$-functions.