The Austrian recovered from losing the second set to record a 6-1 4-6 6-4 6-4 win over Benout Paire.
He was struggling with a shoulder injury but insisted it will not hamper him when he faces Goffin in the next round.
Paire barred from Olympics for 'deeply inappropriate behaviour'
Highlights: Dominic Thiem v Benoit Paire
Meanwhile, the Belgian easily overcame Ivo Karlovic with a 6-3 6-2 6-4 win. The No 11 seed, who was the catalyst behind his country's run to the Davis Cup final in 2015, demonstrated his love of hard courts with a superb display of baseline hitting.
Highlights: David Goffin v Ivo Karlovic
'Very challenging' - Djokovic: Most players experiencing mental and motivational issues
'I fell into a hole' - Thiem opens up on difficult start to season