Boulter said she is is expecting “an absolute battle” against compatriot Dart.
World number 94 Dart did not lose serve against Bai, breaking the 21-year-old in the 10th game to take the first set, and only a rain interruption delayed her progress.
“Playing a Brit on the UK, on the grass, is never an easy draw,” added Boulter.
Dart was one of 11 Britons in action on Tuesday after Andy Murray announced he would not play singles following his back operation.
British number 13 Jacob Fearnley battled to victory over Alejandro Moro Canas to set up a second-round encounter with seven-time champion Novak Djokovic.
The 22-year-old from Scotland, ranked 277 in the world, defeated the Spaniard 7-5 6-4 7-6 (14-12).
Paul Jubb missed out on a first Wimbledon success in agonising fashion, falling in five sets to big-hitting Brazilian Thiago Seyboth Wild.
Jubb led by two sets to love and had a match point in the third-set tie-break, but he could not take it and Seyboth Wild fought back to win 1-6 3-6 7-6 (8-6) 6-4 7-5.
Jan Choinski also went down in five sets, losing 7-5 4-6 2-6 7-5 6-2 to Italian Luciano Darderi.
British wildcard Billy Harris’ Grand Slam debut ended in defeat against Spain’s Jaume Munar, losing 6-4 6-4 3-6 6-3.
Jack Draper – the nation’s leading men’s player – faces Sweden’s Elias Ymer last on Centre Court.
Cameron Norrie, who reached the semi-finals in 2022, is also in action, as too are Dan Evans, Henry Searle and Fran Jones.
