Phase I dose-finding trials are increasingly challenging as the relationship between efficacy and toxicity of new compounds (or combination of them) becomes more complex. Despite this, most commonly used methods in practice focus on identifying a Maximum Tolerated Dose (MTD) by learning only from toxicity events. We present a novel adaptive clinical trial methodology, called Safe Efficacy Exploration Dose Allocation (SEEDA), that aims at maximizing the cumulative efficacies while satisfying the toxicity safety constraint with high probability. We evaluate performance objectives that have operational meanings in practical clinical trials, including cumulative efficacy, recommendation/allocation success probabilities, toxicity violation probability, and sample efficiency. An extended SEEDA-Plateau algorithm that is tailored for the increase-then-plateau efficacy behavior of molecularly targeted agents (MTA) is also presented. Through numerical experiments with both synthetic and real-world datasets, we show that SEEDA outperforms state-of-the-art clinical trial designs by finding the optimal dose with higher success rate and fewer patients.