The non-orthogonal multiple access (NOMA) allows one subcarrier to be allocated to more than one user at the same time in an orthogonal frequency division multiplexing (OFDM) system. NOMA is a promising technique to provide high throughput due to frequency reuse within a cell. In this paper, a novel interference cancellation (IC) technique is proposed for asynchronous NOMA systems. The proposed IC technique exploits a triangular pattern to perform the IC from all interfering users for the desired user. The bit error rate (BER) and capacity performance analysis of an uplink NOMA system with the proposed IC technique is presented, along with the comparison to orthogonal frequency division multiple access (OFDMA) systems. The numerical and simulation results show that the NOMA with the proposed asynchronous IC technique outperforms the OFDMA. It is also shown that employing iterative IC provides significant performance gain for NOMA and the number of required iterations depends on the modulation level and the detection method. With hard-decision, two iterations are sufficient, however with soft-decision, two iterations are enough only for low modulation level, and more iterations are desirable for high modulation level.