sed - Get only the replaced string from a multiline input & omit unmatched lines!

This sed command does what your combination of egrep and sed does:

echo -e "Bla\nBla\nImportant1: One \nBla\nImportant2: Two\nBla\nBla"
| sed -n -e "s/^Important1: *\b\(.*\)\b */\1/p"

You perform the substitution and only print lines that matched, after the substitution.

EDIT: Following was tested on both Mac & Linux.

You can use sed like this:

echo -e "Bla\nBla\nImportant1: One \nBla\nImportant2: Two\nBla\nBla" | \
   sed -n 's/^Important1: *\([^ ]*\) */\1/p'

OUTPUT:
one

Explanation

sed -n 's/^Important1: *\([^ ]*\) */\1/p'

-n # quiet / silent 

{
  s/^Important1: *\([^\ ]*\) */\1/ # replace "Important1: one " with 1st group i.e. "one"
  p                  # print the replaced text
}