The visitors never recovered from a poor start when early tries from Jodie Broughton down the right and Todd Carney, who stepped his way to the try line, were both converted by Pat Richards. Teenager Lucas Albert then scored his first try for the club, although Richards missed the conversion.
But Rovers steadied the ship and fought back through a converted try by former Dragon Kevin Larroyer, Ken Sio added the extras.
However the Dragons had the final say before the break, scoring two further tries through Louis Anderson and Tony Gigot, the latter score converted by Richards, who added a penalty to make it 28-6 at the break.
Both sides squandered opportunities to score after the interval - but Rovers did narrow the deficit in the 63rd minute, with Joe Wardill scoring in the right corner, although Sio couldn’t convert.
The hosts were further clear when Gigot scored again 10 minutes from time, with Richards converting before Sio scored a consolation effort for the visitors in the last minute, Albert Kelly adding the extras.
“I thought Catalans were really good in the first half, but we found it hard for energy when they had so much good ball in so many good positions,” said Webster afterwards.
“But I’m really proud of our second-half performance we won (the half) 10-8 and to come here under the pressure and the amount of fatigue that we’ve been under was a great effort.”
Webster was keen to praise young winger Wardill, who started for Rovers despite playing for the England Under-19’s the previous evening.
He added: “He had to back up for us which was a great effort from him and sums up the sort of player he is.”
Catalans Dragons: Gigot, Broughton, Horo, Duport, Richards, Carney, Albert, Taylor, Pelissier, Casty, Stewart, Anderson, Baitieri. Subsitutes: Bousquet, Mounis, Bosc, Maria.
Hull K R: Marsh, Sio, Clarkson, Thornley, Wardill, Blair, Kelly, Tilse, Lunt, Walker, Donaldson, Greenwood, Mulhern. Subsitutes: Larroyer, Green, Boudebza, Lawler.
Referee: James Child (RFL).