Please release me!

Japt, 16 11 bytes

¡V«´V+ÂX}'.

Test it online! Input number is 1-indexed.

Based on my JavaScript answer. This takes advantage of one of Japt's most helpful features: splitting one string on another before mapping each item, then joining on that string again after mapping.

Ungolfed and explanation

¡  V«  ´ V+Â X}'.
Um@V&&!--V+~~X}'.
                   Implicit: U = input string, V = input number
Um@           }'.  Split U at periods, then map each item X by this function:
   V&&               If V is 0, return V.
      !--V+~~X       Otherwise, decrement V and return X + !V (increments X iff V was 1).
               '.  Re-join the result with periods.
                   Implicit: output last expression

JavaScript (ES6), 44 42 40 37 bytes

Saved 3 bytes thanks to @Neil

x=>i=>x.replace(/\d+/g,n=>i&&+n+!--i)

Input number is 1-indexed.

Test snippet

f = x=>i=>x.replace(/\d+/g,n=>i&&+n+!--i)

console.log(f("1.0.3")(1))
console.log(f("1.2.3.4.5")(3))
console.log(f("10.0")(1))
console.log(f("1")(8))

Vim 20 25 bytes

I unfortunately realized that it didn't handle the case of updating the last number, so I had to add bytes. This is 1-indexed.

DJA.0@"t.qq2wcw0@qq@qx

TryItOnline

Unprintables:

DJA.^[0@"t.^Aqq2wcw0^[@qq@qx

This takes the arguments in reverse order, as separate lines:

3
1.2.3.4.5

Explanation:

DJ                           # Delete the input argument, and join lines
  A.^[0                      # Add a period to the end
       @"t.                  # Move to the "Copy register"th period
           ^A                # Increment the number under the cursor
             qq       @qq@q  # Until an error
               2w            # Move the cursor forward to the next number
                 cw0^[       # Change the number to 0
                           x # Delete the last '.'