Background: Stellar electron-capture reactions on medium-heavy nuclei are important for many astrophysical phenomena, including core-collapse and thermonuclear supernovæ and neutron stars. Estimates of electron-capture rates rely on accurate estimates of Gamow-Teller strength distributions, which can be extracted from charge-exchange reactions at intermediate beam energies. Measured Gamow-Teller transition strength distributions for stable pf-shell nuclei are reasonably well reproduced by theoretical calculations in the shell model, except for lower mass nuclei where admixtures from the sd shell can become important. Purpose: This paper presents a β+ charge-exchange experiment on Sc45, one of the lightest pf-shell nuclei. The focus was on Gamow-Teller transitions to final states at low excitation energies, which are particularly important for accurate estimations of electron-capture rates at relatively low stellar densities. The experimental results are compared with various theoretical models. Method: The double-differential cross section for the Sc45(t,He3+γ) reaction was measured using the NSCL Coupled-Cyclotron Facility at 115MeV/u. Gamow-Teller contributions to the excitation-energy spectra were extracted by means of a multipole-decomposition analysis. γ rays emitted due to the deexcitation of Ca45 were measured using GRETINA to allow for the extraction of Gamow-Teller strengths from very weak transitions at low excitation energies. Results: Gamow-Teller transition strengths to Ca45 were extracted up to an excitation energy of 20 MeV, and that to the first excited state in Ca45 at 174 keV was extracted from the γ-ray measurement, which, even though weak, is important for the astrophysical applications and dominates under certain stellar conditions. Shell-model calculations performed in the pf shell-model space with the GXPF1A, KB3G, and FPD6 interactions did not reproduce the experimental Gamow-Teller strength distribution, and a calculation using the quasiparticle random phase approximation that is often used in astrophysical simulations also could not reproduce the experimental strength distribution. Conclusions: Theoretical models aimed at describing Gamow-Teller transition strengths from nuclei in the lower pf shell for the purpose of estimating electron-capture rates for astrophysical simulations require further development. The likely cause for the relatively poor performance of the shell-model theory is the influence of intruder configurations from the sd shell. The combination of charge-exchange experiments at intermediate beam energy and high-resolution γ-ray detection provides a powerful technique to identify weak transitions to low-lying final states that are nearly impossible to identify without the coincidences. Identification of these weak low-lying transitions is important for providing accurate electron-capture rates for astrophysical simulations.