One of the simplest extensions of the Standard Model (SM) is the two-Higgs-doublet model (2HDM), which contains two neutral Higgs bosons, in addition to a 125 GeV one, and a charged pair. At the Large Hadron Collider (LHC), gluon-induced processes are generally the most important modes for the resonant production of the SM-like Higgs boson as well as its pair-production, and it is generally considered to be the case also for an additional neutral Higgs boson possibly existing in nature. We show that for certain parameter configurations in the Type-I 2HDM, electroweak pair-production of the neutral Higgs states can dominate over the QCD-initiated production. Moreover, it is possible for the pair-production of the charged Higgs state along with a neutral one, which can only take place electroweakly, to have a substantial cross section. We delineate such 2HDM parameter space regions through its comprehensive numerical scanning, requiring their consistency with the most relevant theoretical and experimental constraints. We also highlight some specific di-Higgs signatures that can be probed at the LHC in order to establish the Type-I 2HDM as the underlying new physics model.