You can measure the differences between colors in different ways, depending on the color model/space you're using. Hex colors use RGB, but the HSL model will be more useful here. It records a color as 3 values: Hue, Saturation and Lightness.
We can use SASS's HSL functions to work out how each value of the color differs, and then selectively apply that difference to other colors:
$start-color: #1E95EF
$end-color: #1988DD
//We won't need the hue-difference in practice
$hue-difference: hue($start-color) - hue($end-color)
//These express the difference as a percentage
$saturation-difference: saturation($start-color) - saturation($end-color)
$lightness-difference: lightness($start-color) - lightness($end-color)
@function color-adjust($base-color)
//Apply the lightness and saturation changes but keep the original Hue
@return hsl(hue($base-color), saturation($base-color) + $saturation-difference, lightness($base-color) + $lightness-difference)
//To find the adjusted color, just pass whatever base color you like to the color-adjust function:
background: color-adjust(red)
Here's a demo with a few colors: http://codepen.io/anon/pen/bqOaZZ
Obviously that code can be compacted a lot - I'm thinking that what you really want is to define your variants as a set of Saturation/Lightness changes, and then apply them to your whole palette (rather than trying to 'reverse-engineer' it from one example pair). If you want help with something to do that, let me know!