Raducanu got an unexpected boost when Russian 22nd seed Ekaterina Alexandrova, her scheduled opponent, withdrew because of illness on the morning of the match with Zarazua, a loser in qualifying, taking her place.
With former England icon David Beckham watching on from the Royal Box, Raducanu looked nervous early on against an unorthodox opponent but eventually contained the unforced errors that littered the first set to move into round two.
It was a welcome win for the former US Open champion who missed last year's Wimbledon because of injury and who has struggled to reach the dizzy heights of her fairytale in Flushing Meadows in 2021.
It was also a welcome win for the home nation after the first four British players in action on Monday all lost - adding to the tale of woe at the French Open where all six British singles players suffered first-round defeats.
After a scrappy opening to the match, Raducanu appeared to have settled down when she led 4-2 but Zarazua, ranked 98th in the world, shot back to win three games in a row and Raducanu twice had to hold serve to stay in the first set.
Raducanu, battling back up the rankings after missing the second half of 2023 because of ankle and hand surgery, blazed through the tiebreak and looked far more comfortable after that to set up a clash with Belgium's Elise Mertens.
"I found out at 10:30 today (that Alexandrova had pulled out) and for the last three days we were playing with similar opponents. It took a little bit of adjusting to find my feet," Raducanu said on court.
"It was an incredibly difficult match. I've barely played a defender who's landed the ball on the baseline and it took a lot of strength to get the ball back."
Nineteen British players were in the men's and women's singles but earlier in the day there were defeats for Charles Broom against three-times Grand Slam champion Stan Wawrinka, Heather Watson, Liam Broady and Arthur Fery.
British wildcard Yuriko Miyazaki followed Raducanu's victory as she notched her first Wimbledon win, thrashing Tamara Korpatsch 6-2, 6-1.
