The Frenchman dominated the first set, breaking Soeda three times before taking it 6-0 in just 21 minutes.
Soeda improved in the second set, winning two games off the Gasquet serve, but he still found it tough to cope with the number eight seed, who again broke on three occasions before serving out the set 6-3.
The third set was a much closer affair, with both players giving little away off their own serve. Neither could be separated as the set entered a tie-break, which Soeda claimed 7-5.
In the fourth set both players struggled to dominate on their own serves, with both breaking twice. However, Gasquet made an important third break in the eighth game to move 5-3 ahead, before holding once more to secure victory.
He will now face Bernard Tomic in the third round.