Welch, who led GE for two decades, died Sunday at age 84, his wife, Suzy, announced Monday.
"As we look around the company today, his memory and his influence certainly loom very large for us," Culp said on "Squawk on the Street."
"Jack was first and foremost about talent and the team, and he had a deep passion for winning, and those are very much values that we embody today," said Culp, who became CEO of GE in October 2018
"The greatest tribute we can pay Jack going forward … is to continue to strengthen this company with an eye toward winning," said Culp, who is in the midst of trying to turn around GE, which has fallen on hard times since the Welch era.
Welch grew GE's market cap from $12 billion to $410 billion, making it one of the world's most valuable companies, but it fell on tough times after his departure in 2001.
The industrial conglomerate struggled during an era that included the dot-com bust and Great Recession, and some of its components — such as NBCUniversal and GE Capital — were either sold or spun off into separate entities.
Welch started at GE in 1960 as a chemical engineer and eventually became chairman and CEO in 1981.
Despite no overlap in their tenures at GE, Culp said the two had crossed paths. Their first meeting came in the late 1980s, when Culp was a student at Harvard Business School.
"I remember it vividly. I suspect Jack didn't," Culp said. "But anyone who grew up in the industrial world, or even more broadly during that time, I think went to school on GE, whether it was an orientation toward talent, globalization and the like, those were ideas that Jack was able to execute on and I think taught many of us how to do the same."
Welch also had a direct question for Culp, shortly after Culp took the reins at the embattled industrial giant.
"I remember when I saw him last, what I'll always remember most vividly, was Jack leaning across the table and asking, 'So how exactly are you running the company?'" Culp said. "Jack was still very much in the game, very much committed to our success. Frankly, to have Jack ask me how I'm running GE was a rather humbling moment."