Yes, there is a transfer switch (relay) that will choose shore power when it is available or inverter power when it is not, assuming the inverter is enabled. Not sure how your unit is wired but often there is a main panel that gets power from shore or generator and from that will be a circuit feeding the inverter. The inverter in turn will feed a sub panel which will have most, if not all, wall outlets and microwave. This can vary from brand to brand however. A/C units, water heaters, etc. will be fed from main panel using only shore or generator power.
2013 Winnebago Aspect 27K
E450, V10, CRV, Blue Ox