This might seem similar, but in reality I don't think you can draw any parallels here. If you do - do it carefully.
Here, за- doesn't mean the beginning, quite often it means the opposite - завершить, закончить.
What concerns the English equivalent of re- often you can use пере- (like in to be reborn = переродиться, to rebuild = перестроить), but not always.
Sometimes you cannot even express re+verb in one word, instead you will use the adverb заново, снова or повторно (reenter - войти повторно, redo - сделать снова or переделать).
As far as I can tell, there's no 'universal' prefix that will work here.
Finally - the 'over-' part. Usually you can use the prefix пере- (to overheat - перегреть, to overwork - переработать, to overload - перегрузить), but sometime you won't be able to express it in one word like in over-react - for a literal translation you will need 3 words - среагировать излишне (чрезмерно) сильно (to react with excessive strength).