Frenchman Benneteau got off to a shaky start and had to save three break points in the opening game before holding serve.
The next six games went with serve until Benneteau broke to take a 5-3 lead, serving out the 37-minute set to love in the next game.
Janowicz responded in the second, rattling off four games in a row to take a 5-1 lead. Benneteau pulled one break back but couldn't draw level in the set as Janowicz came through 6-3.
The Pole then broke at the first time of asking in the deciding third, but his serve deserted him soon after as Benneteau broke twice in a row to take a 4-2 lead.
Benneteau was also guilty of failing to take advantage as Janowicz broke right back. Holds of serve were traded in the next three games before Janowicz broke in the 11th game and then served out the match to love.
He will face either Frank Dancevic or Lu Hen-Ysun in the second round.