Flipkens declined to talk after the match about the terrorist attacks that took place in Belgium on Tuesday.
The 504th-ranked Robson is playing only her second WTA-level tournament of the year, losing in the first round of the BNP Paribas Open.
The left-handed Robson, who has been dealing with a left wrist injury for two years that required surgery, is entering tournaments via a protected ranking.
The 22-year-old attained a career-high ranking of No. 27 in 2013.
Americans Madison Brengle and Nicole Gibbs also advanced to the second round by ousting higher-ranked players.
The 60th-ranked Brengle defeated 43rd-ranked Camila Giorgi of Italy 6-4, 6-3, while the 74th-ranked Gibbs ousted 58th-ranked Yulia Putintseva of Kazakhstan.
Only 12 women's main draw matches were on the opening day schedule at the Miami Open.
Play wasn't even scheduled on the stadium court which was left open for the top players to practice.
Players such as Rafael Nadal, who reached four finals here in 11 visits to the tournament, drew a crowd to the stadium when he practiced in the early evening.