The Englishman, who is from St Albans, competes on the US collegiate circuit and has one year left on his studies at the University of San Diego.
Most college sports in America are strictly amateur, so Tarvet will be unable to claim the vast majority of his £66,000 prize money for reaching the Wimbledon first round.
“There’s a lot of emotions, but the the main one is just happiness,” Tavert said.
“Ever since I was a little kid, it’s been my dream.”
Tavert previously defeated top-250 players Terence Atmane of France and Canadian Alexis Galarneau in the first two rounds of qualifying.
Earlier on Thursday, British world number 550 Hamish Stewart fell 6-3 4-6 6-3 6-4 to Swiss Leandro Riedi.
Emily Appleton also missed out on a main-draw spot, losing 6-2 2-6 6-0 to Veronika Erjavec of Slovenia.