The commonly used energy balance model from Gregory et al. (2002) that underlies many published estimates of Equilibrium Climate Sensitivity (ECS) and Transient Climate Response (TCR) to anthropogenic forcing requires only four parameters for calculation of ECS and three for TCR. Both estimates require a value for the increase in global mean surface air temperature (ΔT) over a period of time, the increment in forcing driving the temperature change over that period (ΔF), and knowledge of the radiative forcing resulting from a doubling in CO2 concentration (F2×CO2). For ECS a value for the associated global heating rate (ΔQ) is also required. Each of these parameters has a best estimate available from the IPCC’s Fifth Assessment Report, but the authors did not provide best estimates for ECS and TCR within the broad uncertainty ranges quoted, 1.5–4.5 K for ECS and 1.0–2.5 K for TCR. Best estimates for ECS and TCR consistent with AR5 best estimates for ΔF and F2×CO2 are provided here. A well-known heuristic model was modified and applied to seven observation-based global temperature datasets to isolate temperature trend due to anthropogenic forcing from confounding effects of variability due to volcanism, cycles in solar irradiance and internal climate variability. The seven estimates of ECS and TCR were remarkably similar despite very large differences in time-base of the datasets analysed, yielding best estimates of 2.36 ± 0.13 K and 1.58 ± 0.09 K respectively at 95% confidence based on the AR5 best estimates for ΔF, F2×CO2 and ΔQ from Wijffels et al. (2016). The ECS and TCR best estimates here are tied to those AR5 and ΔQ best estimates, but can be simply scaled were those best estimate values to be refined in the future.