The former world number one had fallen at the first hurdle in six of his previous nine appearances at the claycourt Grand Slam, highlighting his uneasy relationship with Roland Garros and the sport's slowest surface.
Medvedev showed early signs of frustration when he dropped serve and then allowed Walton to build a 4-2 lead in the opening set, which the 30-year-old relinquished with a forehand that sailed over the baseline.
The recovery was swift and emphatic as Medvedev regained his rhythm to wrest control by claiming the next set for the loss of only one game, but the sixth seed could not maintain his grip and let the third set slip away.
World number 97 Walton, who was seeking a first win over a top-10 player, surrendered the fourth set but fought on bravely in the decider to break back at 4-4, before producing a tight hold and then dismissing Medvedev for a famous victory.
Up next for the 27-year-old is a second-round meeting with American Zachary Svajda, who beat Australian Alexei Popyrin.