It all started so well for the visitors when a short Robins goalline drop-out fell straight to Bennison, who darted back inside and under the posts.
But the hosts’ response was swift as Lewis created space for Davies to sprint into the right corner.
Davies was denied a second moments later when his flying one-handed finish was ruled out by video referee Cameron Worsley for obstruction.
The Robins were far from their best but hit the front after 24 minutes when the ball was quickly shifted out to Lewis, who brushed off a challenge and dived over for his 15th Super League try of the season.
Powerhouse prop Tevita Pangai Junior made an instant impact on debut when his first carry off the bench gained ground before Lewis sent winger Burgess skipping away to the left corner to score.
York battled to prevent further damage before the break, but conceded again when Peta Hiku unselfishly teed up Batchelor to drop over the line.
A second acrobatic one-handed finish from Davies this time counted as the hosts extended their lead soon after the restart.
The Robins temporarily lost their spark when Lewis needed treatment after he was pulled down awkwardly on his knee before playing on.
But the impressive Tyrone May got the ball rolling again, breaking through a tackle and sending Broadbent under the sticks before Martin finished a flowing try from one edge to the other.
Hull KR: Broadbent; Davies, Hiku, Gildart, Burgess; Lewis, May; Hadley, Lawton, Amone, Martin, Batchelor, Minchella.
Interchanges: Litten, Luckley, Dezaria, Pangai.
York: Mata’afa; Bennison, Buchanan, Wood, Galeano; Hunter, Harris; Martin, Inman, Vaughan, Field, Reynolds, Thompson.
Interchanges: Balmforth, Vuniyayawa, Va’a, Hingano.
Referee: James Vella.