The 23-year-old, who climbed to 45th in the latest ATP world rankings, lost serve at the start of both sets before losing the final three games.
Vinolas, 31, is ranked 38 places below the Briton and will now face Russia's Daniil Medvedev in the last 32.
Norrie is set to play at next month's French Open at Roland Garros.
The loss comes less than a week after Norrie exited the Monte Carlo Masters in the round of 16 to Italian Lorenzo Sonego, ranked 40 places below him.
In the second round on Tuesday, world number three Alexander Zverev was defeated in three sets by Chile's Nicolas Jarry, who qualified for the tournament as a lucky loser.
Jarry, 23, saved a match point on his way to a 3-6 7-5 7-6 win, inflicting a fifth defeat in seven meetings on his German opponent.
Third seed and world number five Dominic Thiem progressed to the third round with a 6-3 6-3 victory over Argentine Diego Schwartzman, while two-time Barcelona champion Kei Nishikori recovered from 4-1 down in the first set to record a 7-5 6-2 win over 21-year-old American Taylor Fritz,
Spain's David Ferrer, who will retire after next month's Madrid Open, also advanced following a 6-3 6-1 win against Mischa Zverev.
Defending champion Rafael Nadal, chasing a record 12th Barcelona Open triumph, begins his tournament against Leonardo Mayer Wednesday.