As many wise folk have said, to my mind there are essentially 3 options when installing an uprated engine:
A) uprated engine, but keep the same drivetrain/rotors, but be able to apply more torque (turning force) to keep them turning at steeper pitch angle (more drag) at the same RPM. This enables an increase in overall lifting capacity but assumes that the drivetrain is capable of handling an increase in torque, the aerodynamics of the rotor blade work sensibly at an increased pitch and the fuselage is capable of coping with increased mass.
B) Uprated engine used to overcome a limiting factor, but not used to increase normal torque. For example, your engine might be limited at high temps/density altitudes by outlet/turbine temperature, so changing the hot components of the engine for improved parts might allow a higher rated temperature, meaning you are able to achieve the same torque in worse environments, thereby improving lift in those without changing any other components (because you're not really changing the max rated torque of the drivetrain or pitch angle of the blades).
C) change everything as part of a redesign - obviously gives improved lift. New blade profiles, transmissions and engines = step change in performance but significant cost.
Case B) often also applied to twin engine helicopters where improved engine performance doesn't increase twin engine lift but does increase single engine failure (OEI) safety margins, which can in turn lead to an apparent increase in overall performance due to regulations allowing it, rather than the aircraft being physically capable of it.
Case A) has been known to apply where the original aircraft was "over-designed" for the original engines (e.g. engine manufacturer failed to deliver promised performance from shiny new wonder-engine, but the airframe was designed for it or where it was known that a new engine was likely early in the airframe life.)