Multiplying array elements with Javascript

First, those values are strings, not numbers, so the first thing to do is make them numbers:

 array.map(Number)

Then multiply with .reduce

 array.map(Number).reduce(function(product, value) { return product * value; });

edit — a comment wisely notes that the * operator will try to convert its operands to numbers anyway. With the explicit conversion to numbers, you could add a filter to see if there were any NaN values produced, but NaN spreads like cancer so it really wouldn't matter. Consider the numeric conversion optional.

Now, in general, if I were getting an array of numbers-as-strings from an API, I'd be inclined to do an explicit conversion, because in a sense that means that there's really something wrong with the design of the API. I'd prefer to isolate any "interesting" code from weird stuff like that.

Fast and efficient, go with a good old for loop, see http://jsperf.com/reduce-v-for-javascript

The fastest form of loop is standard for, See map vs for-loop performance test.

NOTE : The slower is map() function like you can see in test.

var Array = ["1", "2", "3", "4"];
var total=1;

for (var i = 0; i < Array.length; ++i) {
    total *= Array[i];
}

alert(total);

Hope this helps.

The most efficient/performant way will be always to use regular loop:

var array = ['1', '2', '3', '4'], i = array.length, result = 1;
while (i > 0)
  result *= array[--i];

But since that kind of performance will never matter in the real world, I suggest you to use reduce:

var array = ['1', '2', '3', '4'];

var result = array.reduce(function(a, b) {
  return a * b;
});

console.log(result);

You can use reduce

var sum = array.reduce(function(pv, cv) { return pv + cv; }, 0);

with arrow functions in ES6, it is even simpler.

sum = array.reduce((pv, cv) => pv+cv, 0);

or with for loop

var total = 0;
for (var i = 0, n = array.length; i < n; ++i){
 total += array[i];
}

and with crazy way

var a = [1,2,3,4,5];
sum = eval(a.join("+"));