The number 12 seeds are hoping to go at least one better than their semi-final exit in the Australian Open earlier this year and came out of the blocks quickly today, breaking serve in the third game of the match.
But the Frenchmen responded well and had two break points in the very next game, but Goransson and Verbeek defended well to open a 3-1 lead in the first set.
The next four games all went with serve to give the Swedish-Dutch pairing a 5-3 lead, and they came very close to forcing another break off Muller's serve in the ninth game, only for the Frenchmen to save two set points.
However, a comfortable hold in the next game wrapped up a 6-4 first set win for Goransson and Verbeek.
Despite low doubles rankings, Mpetshi Perricard and Muller are both ranked inside the top 50 in the men's singles, and they proved tough for the two 31-year-olds to break down as the first 10 games all went with serve, with none even reaching deuce.
The following two games were no different, meaning the second set went to a tiebreak.
Goransson and Verbeek twice broke their opponents to earn 3-2 and 5-4 leads in the tiebreak, but on both occasions their opponents broke back soon after, saving two match points in the process.
Finally, the Finn and Dutchman forced a third break of serve to go 7-6 ahead, then finally secured their passage to the Second Round with an 8-6 win in the tiebreak, and the straight sets victory (6-4, 7-6).
In the next round, Goransson and Verbeek will face either Nuno Borges and Arthur Rinderknech, or Zizou Bergs and Jesper de Jong.
Goransson will return to the Roland-Garros courts tomorrow as he plays alongside the Russian Irina Khromacheva in the mixed doubles against Brazil's Luisa Stefani and Argentina's Maximo Gonzalez