Question: As per power equation $P=V×I$, voltage is inversely proportional to current as power of load remains constant. So why we get greater output from electrical loads with increase in voltage. (whether it may be resisitive or inductive)?

Answer: 
The question suggests to me that you have been confusing yourself.

Usually, when you increase the voltage to a load, the current will increase accordingly. The result is that the power delivered (which, as you stated, is $P=V\cdot I$), will increase.

You cannot simply say "now I pretend the power is constant, and I increase the voltage, so why isn't the power constant?". You would have to ask yourself "what happens to the current?". And of course, if the construction of the load doesn't change, then increasing the voltage will increase the current. That's Ohm's law.

The only "violation" that occurs is in your initial statement, where you posit that power will be constant. Starting from a false premise you end up in a confusing pickle.
There’s no violation of ohms law if we write:

(1) $V=P/I$

All this says is that in order for $P$ to be constant if voltage is increased than current must be decreased. E.g., if we double the voltage we need to halve the current for $P$ to be the same. 

Per Ohms Law:

(2) $V=IR$ 

Plugging value of $V$ from (2) into (1):

$IR=P/I$

$P=I^2R$