Jesse Zgraggen also scored for the Royals (7-4-0).
Rourke Chartier and Myles Bell each had a goal for the Rockets (4-5-1).
Coleman Vollrath made 30 saves in his first regular season appearance for the win while Jordon Cooke turned aside 19 shots in a losing effort.
Chartier opened the scoring for the Rockets at 6:53 of the first period, tipping home a shot from the point by defenceman Riley Stadel.
Walker answered for the Royals in the second period with his third goal of the season to even the score at one apiece.
Bell would reclaim the lead for the Rockets at 19:26 of the second period on a five-on-three power play. Bell scored his sixth of the season on a one-timer from the top of the face-off circle.
Zgraggen capitalized for the Royals just as the ensuing power play from a Ryan Olsen misconduct penalty expired.
Walker scored his second goal of the game on the power play to win it for the Royals at 17:44 of the third period.
Both clubs went 1-for-5 on the power play.