A typical goal of a quantum simulation is to find the energy levels and eigenstates of a given Hamiltonian. This can be realized by adiabatically varying the system control parameters to steer an initial eigenstate into the eigenstate of the target Hamiltonian. Such an adiabatic quantum simulation is demonstrated by directly implementing a controllable and smoothly varying Hamiltonian in the rotating frame of two superconducting qubits, including longitudinal and transverse fields and iswap-type two-qubit interactions. The evolution of each eigenstate is tracked using time-resolved state tomography. The energy gaps between instantaneous eigenstates are chosen such that, depending on the energy transition rate, either diabatic or adiabatic passages are observed in the measured energies and correlators. Errors in the obtained energy values induced by finite T1 and T2 times of the qubits are mitigated by extrapolation to short protocol times.