Thanks for defining what you mean by exploitation. Now, exploitation can only exist in a market where workers have no choice but to accept the conditions of a single employer. This is typically the case when you are a slave for someone else and you have lost your freedom completely.
However, the have-nots (who are willing to work) can probably find several occupations, from different employers. They will have some range of choice concerning the salaries, the conditions and so on. Maybe not a large choice, but some choice. And one thing is clear: the less minimum wage there is, the more choice of occupations there will be, since there will be more employers generating occupations where profit can be made, since you lower the costs of labor at first.
Those occupations would not exist if it is easier or more profitable for people to stay on welfare.
And as it has been demonstrated in so many cases/countries, the longer one stays on welfare, the most likely one is not going to look for a job, welfare is actually a promoter of misery in the long term, especially for those who are at the very bottom, since they get no real incentive of getting out of it.
In a sense, politicians are guilty of exploitation as well, based on your definition. They take advantage of some voters' misery to promise them continuous welfare for which they are not responsible to pay for, and they further their own goals by getting rich in the process.
But, to come back on "exploitation" again. You are considering that is is a zero-sum game, basically. One, the exploiter, is gaining something while the exploited is losing. But economics almost never work that way. Both employer and employee have something to gain out of it. Both profit from the situation. The Employer makes money, can extend his business and ends up needing more people in his payroll. The Employee makes enough money to survive, has an occupation and has a social activity within society - they build relationship/reputation and somehow experience. They may not be making much monetary profit, but they are still gaining something out of it.
Please do not forget that all people who lived before us had to go through misery before reaching nowadays' living standards. If you come back a hundred years ago, then you may well consider that everyone then was being "exploited", but that is not how they would see themselves at the time.