One of the ways they made this work was by assigning temporary powers to elected/appointed individuals appropriate and sufficient to get their job done, whether that was provincial governor, tribunus plebis, consul or whatever. A notable title was that of "dictator" (where the modern word derives from) which gave sweeping temporary powers to a general sufficient to execute a war.
Julius Caesar famously abused this system, effectively becoming the first emperor (despite no such position existing) by assuming the title of "dictator perpetuo" (perpetual dictator), although that did not end well for him. His successor, Augustus (whose adopted name later became the name of the office of emperor) initiated a more acceptable version of this by simultaneously holding multiple "elected" positions thereby having the power of them all.