Dockett and Dansby played a key role in helping the Cardinals to the number one defence against the run last season, but the latter has since left the team to join Cleveland on a four-year, $24m (£14.3m) deal.
Dockett believes that his side are close to building a team that can challenge for the Super Bowl and suggested that Dansby's decision to jump ship could cost him a championship ring.
"He chased the money. I got a lot of respect for our guy that left. I love him like a brother, but we was one or two pieces away from really making a lot of noise. But we're going to regroup. I personally feel like he chased the money versus chasing a ring," Dockett told reporters.
"No knock toward Cleveland. I don't want people to think I'm saying Cleveland don't have a chance. Everyone has a chance. But I just felt like it was made for him to be here. Again, the financial part is totally different. I don't know anything about that."
The 32-year-old also hinted that he felt let down by Dansby's decision and vowed to use his departure as motivation to succeed next season.
"When you look at everything we've done this year, the sacrifices our defensive line made for that certain individual to make his plays, and go into a game not being selfish, we don't care, our defensive line don't care about sacks and tackles and stuff," he added.
"We're coming to do a job, to hold guys off our linebacker so we can be the number one run defence. I just think that our opportunity was to be right there and to really get into the playoffs and make a run for it.
"We're not going to complain about spilled milk. I'm going to wish him well in Cleveland, and we'll see who's where at the end of the season. That's our goal, and that's our motivation."
The Cardinals will begin the 2014 season with a home match against the San Diego Chargers on September 8.