Katarina didn’t want Red to know that Liz had betrayed him for her and that is a very maternal thing to do and it is something that I think more than any other thing that had happened prior convinced Liz to put her eggs in Katarina’s basket.
Liz understands that Katarina is a very dangerous spy and someone who is not the most trustworthy person.
However, in the final episode, she did something that genuinely moved Liz and I think motivated her to align herself with Katarina in a very real way.
She in some way sacrificed herself for Liz by giving Red information that allows him to track her down.