The first rookie to appear on the scoresheets this season, at the 15th attempt, Gutierrez made his breakthrough with seventh place at Suzuka on team founder Peter Sauber's 70th birthday.
"It is such a big relief because we've been seeing how he has been improving over the last few races and he's had a really rough start into the season," Kaltenborn told Reuters.
"It just takes that little bit more to let something explode or just open up in him. And that's what's happened now," she added.
The 22-year-old was handed his opportunity after compatriot Sergio Perez left Swiss-based Sauber for McLaren at the end of last year and many saw his promotion as more of a reflection of the team's desire to retain key Mexican sponsors.
His performances in the early races were ragged, with qualifying difficult and German team mate Nico Hulkenberg in a different league. His future at the team remains uncertain.
In Japan, the Mexican qualified 14th and then moved steadily through the field.
"I think it's very important for him because he has been criticised quite a lot," Kaltenborn said.
"When we took him up, we were convinced and we still are that he is a very talented driver.
"When you have a rookie you always have a certain risk attached to it and it was no different with him. We knew he will make mistakes and it was quite a lot at the beginning but that's all part of growing into Formula One."
Gutierrez had out-qualified Hulkenberg for the first time in Singapore, when he also made his first appearance in the final phase of qualifying.
He failed to convert that into points but made no mistake on Sunday.
"Finally I can feel great satisfaction," he said. "It has been a lot of work to get up to this point... over the past few races we have been improving a lot and we have to keep optimistic."
(Editing by John O'Brien)
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