I agree with that. However, nothing is perfectly inelastic, and so increasing the minimum wage would still increase unemployment, even if only by a little amount.

Then let's assume it's just highly inelastic. If raising the minimum wage causes a small decrease in employment, but the raise is high enough that there is still an increase in the total wages for all workers in the economy, that puts more money in the hands of consumers, which potentially results in companies needing to hire more workers to meet the demand resulting in a overall increase in employment.

It also creates a countervailing force that increases employment: people on minimum wage spend their money which creates demand, which creates minimum wage jobs.