The Bulls started well and led 6-0 early on, but Rovers fought back to enter the break 24-6 ahead.
It was a similar story after half-time when Bradford scored two tries to reduce the deficit to eight points, before a second touchdown from Featherstone striker James Harrison put an end to his retaliation and the Rovers l ‘win 41-16.
Webster considered the Betfred Challenge Cup first round draw to be much closer than the score suggests and described it as a “tough competition”.
He said: “I thought Bradford was very committed and they made few mistakes at the start which was in total opposition to us – we made thousands of mistakes and made it very difficult. for ourselves.
“We have good players who can break tackles and score tries, but once we learn to respect the ball a little more, we’ll be a lot better.
“We had to do really well to keep Bradford on the outside and easily could have been further behind at one point.
“I’m very happy to be in the next round, but I hope we can improve in a few areas.”
Harrison, who played four games on loan for Leeds Rhinos last year, was among the Rovers substitutes and Webster praised the impact of his bench.
Featherstone Rovers 41 Bradford Bulls 16 – James Harrison wins the Challenge Cup …
“Dean Parata was good when he came in,” he reflected.
“James Harrison was exceptional and Dale Ferguson came in and created a little more tempo, but if I was [Bulls coach] John Kear, I would be proud of my players – I thought they played well.
“Sometimes we’ve come up with really good games from really good people.
“It wasn’t that we continually broke Bradford, we just had players who could break a tackle and come up with plays.”
The Rovers’ next two games are both at home against the Batley Bulldogs, in the second round of the Cup this weekend before the teams meet in their first Betfred Championship game on Friday April 2.
The Rovers will likely be without Ferguson for both games after sustaining a hamstring injury on Sunday.