The loss ends any chance of Manly making the finals this year under first-season coach Trent Barrett.
Manly skipper Jamie Lyon missed an 80th-minute penalty goal, from a Sam Kasiano ruck penalty, that would have kept the visitors' slim playoff hopes intact.
The Sea Eagles were the better side for much of the contest but missed a number of field-goal attempts in extra time as the Dogs held their composure.
However, Canterbury's effort will have only given ammunition to those who have written them off as a premiership threat.
Cherry-Evans chipped and chased for himself and outleapt Will Hopoate for the game's first try in just the fourth minute.
Brett Morris scored his first try from a "Falcon assist" from his brother Josh when the Steeden bounced off his head and Brett pounced in the in-goal.
Brett added his second five minutes later, scything his way past three defenders to score his 130th career try.
The Dogs led 12-6 at the break before Matt Wright crossed two minutes after halftime on the back of a Lyon linebreak.
The Dogs went ahead again in the 56th minute through winger Perrett but his opposite Jorge Taufua answered six minutes later after a Nathan Green tapdown.
Taufua was denied 10 minutes later via a double movement and the game moved into the golden-point period after Lyon's miss.
Manly centre Dylan Walker suffered a dislocated shoulder in the first half and could have played his last game of the year.
The win was the Bulldogs' seventh in eight games and their eighth in nine golden-point matches and will hold their spot in the top four regardless of other results in round 23.
"It was a pretty scrappy affair," Bulldogs coach Des Hasler said of his 200th NRL win with the clipboard.
"The positive for us was (winning after) the five-day turnaround.
"It was against a side that was desperate. Their season was on the line - you could see that.
"We came up with some great defence.
"They are (ugly) wins, but we have to have them. There was a lot at stake in this game.
"The fact that we can graft out the win and overcome moments of adversity to hang in there and keep turning up, that has to be a good sign."
Manly are on 20 points with three matches to play meaning they can't make the 28-point cut for the final eight.
"The boys are shattered," coach Barrett said.
"From my point of view, I couldn't be more proud of the effort and fight. We knew we had to win to make the finals and we gave ourselves every chance.
"We had the opportunity to win the game there in golden point but it wasn't to be."
- AAP
