Depending upon what the break did (if it ripped a trace up and took out a line you need then you get to fix that as well) but yeah voltage is voltage for this sort of thing -- if there is a current path then you get the current flow and it is good again. I will note though that sometimes these sorts of fixes can make a later fix a bit harder.
The trouble will come if you want to use those rechargeable batteries/battery packs, much less from a different device. Rechargeable batteries these days are not always just a matter of "apply voltage, hurry up and wait" and different chemistries (NiMH and the various flavours of lithium these days being what you are most likely to meet) and different devices have different approaches (some batteries have onboard management, some will want said management in the device, some devices will come with a full management* and others will be temperature only). If it is the same voltage (amp hour rating/capacity does not matter so much as long as it is not less than about a 1/3 of the original, though obviously a 1/3 of the original means about a 1/3 of the charge life if not less) as what is intended and has a compatible management approach. Alternatively if you have a means of charging the battery outside the thing independently (or can wire in a charge management setup as well -- see the various original GBA to lithium stuff people have been doing for a few years now) then you can skip that and wire to whatever points on the PCB normally take the power from plain old dumb AA batteries** and in that case we are back to "if it makes the right voltage and has enough capacity to be useful then eh".
*arguably a wallet too fat error and means of making sure you only buy the official batteries but it is what it is. I don't know what MS have done for the xbone controllers, though they did have some fun on the 360 stuff, the PSP had this (the thing with pandora batteries relying upon it) and it has been common practice in tools for years now.
**indeed if you buy a third party charger pack then it will likely replace the AAs and instead get a power lead into that rather than use the onboard charger port. Dodging issues with needing sanctioned chips and whatnot being one of the reasons for that move.
