We prove a new expression for binomial sums with harmonic numbers. Our derivation is based on an alternative argument for the Euler transform of these sums. The findings complement a result of Boyadzhiev. To demonstrate the usefulness of our alternative approach, several examples are discussed. We rediscover some known identities for harmonic numbers and present some new ones. In particular, we derive some new identities involving harmonic numbers, and Fibonacci and Lucas numbers.