The Dons, inspired by veteran Paul Cooke, led 12-0 after 16 minutes and were behind only 28-24 until 15 minutes from time, when Widnes eased away to seal a 42-28 triumph.
Teenager Adam Lawton scored a hat-trick for the Vikings but Betts said: "I am embarrassed and so are the players by what they served up."
He added: "Doncaster came with a good attitude and had a real dig. This game is all about attitude and they had the right attitude. They bust us down the middle and it was a slap in the face for us. We just didn't turn up. We are full-time players and should have been able to win in a canter. I am upset at the application of a few of my players."
Regarding the performance of 19-year-old Lawton, Betts said: "Adam Lawton was Johnny on the spot for two tries and ran a good line for another and just has to keep working hard."
The Dons, bottom of the Championship with two wins from 10 games, were considered huge outsiders to claim an upset win but they gave a fine account of themselves. Coach Tony Miller was proud of his side's performance after seeing them run in five tries and hopes it will help them turn around their league form.
"The players were a credit to themselves and the club," Miller said. "I thought the margin flattered Widnes and we had a great effort from the first minute to the 80th. We dug in and showed how much character there is in the team.
"There were some outstanding performances with Mike Emmett brilliant at loose forward and Brett Waller was the best prop on the field.
"He got us on the front foot, has a good engine and is a handful for any team. I thought there was a forward pass in the build up to their try which made it 18-12 at half-time but we battled to the end.
"It was game on at 28-24 and if we had been able to hold on for another few minutes they may have started panicking. But after this performance there is no need to fear anyone in our division."
- Sports & Recreation