PHP, 81 74 bytes

<?=array_reduce($argv,function($c,$x){return[$x,$c[1]+($x<$c[0])];})[1]+1;

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PHP, 81 74 67 bytes

array_reduce($a,function($c,$x){return[$x,$c[1]+($x<$c[0])];})[1]+1

Implementation of algorithm from ovs' answer: count all pairs where right < left, then add 1.

Since array_reduce takes only one "carry" param, but we want each iteration to know both the running total and the previous value, we make 'carry' be an array with both elements in, which is a few bytes shorter than making a global, but makes a warning the first time we access the (uninitialized) array.

Reduced from first submission by moving all calculation into the return.

Reduced further since it apparently doesn't need to be a program, just a code snippet that takes an array and outputs a value, so all the boilerplate can go away into a header/footer.

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PHP, 81 bytes

<?=array_reduce($argv,function($c,$x){$c[1]+=$x<$c[0];$c[0]=$x;return $c;})[1]+1;

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PHP, 81 74 bytes

<?=array_reduce($argv,function($c,$x){return[$x,$c[1]+($x<$c[0])];})[1]+1;

Try it online!