Having romped through the first set in 37 minutes, Smith trailed 2-4 in the second but then regained control to gain a victory that should help her to improve her present world ranking of 128. McQuillan is ranked 63 and Smith will gain bonus points as well as for the win itself.
"This is one of my best wins ever though I have beaten Rachel before, so right from the start I thought I could win,'' Smith said. "It's a big boost for my confidence and now I just want to win as many matches as possible in the run up to Wimbledon."
Two other British girls, Karen Cross of Devon and Lorna Woodrofffe of Surrey were both beaten in the first round.
On Monday there was no play because of rain and yesterday downpours delayed the start by more than three hours. The top seed, Steffi Graf, who has a first-round bye, practised indoors for 90 minutes but declined to be interviewed.