Odubajo, a free transfer summer recruit, conceded two penalties either side of half-time at Deepdale. He brought down Sean Maguire on both occasions and Daniel Johnson converted the two spot-kicks to take his tally to four goals in three matches.
Although Steven Fletcher handed the Owls a 78th minute lifeline by heading home Adam Reach’s inviting cross, the Lilywhites held on to sentence Bullen’s team to a second consecutive away loss.
Bullen said: “Ultimately two mistakes cost us.
“But like I said to the players in the changing room, one thing we don’t do as team is throw anybody under the bus or leave anybody hung out to dry.
"There is a kid in there that’s really disappointed but we get round him and back him up.
“He (Odubajo) is a good player and he has played in the Premier League so he is a top-class player. He had an off day today."
The Scot felt "there was nothing in the game" up to Johnson's 32nd minute opener.
"I don't think any team had really taken a grip of the game," he said. "They probably shaded possession and we had flashes here and there.
"There was nothing really in it and then we found ourselves 2-0 down.
"We had to chase the game a little bit. We almost got there. We had a go. I have always said we will have a go, no matter, what and the lads had a go so they deserve a lot of credit from that point of view.
"I do't think it was a great spectacle other than the two penalties but it got exciting at the end."