Crichton struck from the tee in the 76th minute at Accor Stadium on Monday after Eels hooker Tallyn Da Silva was penalised for a questionable high shot on Matt Burton.
Burton laid down seeking treatment after Da Silva's contact and failed to finish the match as he was sent for a head injury assessment.
Canterbury's attack was woeful for large parts of Monday's game but that mattered little to Bulldogs coach Cameron Ciraldo, whose 13th-placed side (5-8) kept their faint finals hopes flickering.
"There's clearly a little bit of (a) confidence (shortage), they're the most scrutinised sort of halves pairing in the competition," Ciraldo said.
"Our halfback (Lachlan Galvin) is 20, and has to hear a lot of commentary about how he's going.
"There was just a lot of tension around the whole thing.
"We've had 12 weeks where we have had to work really hard and fight through some adversity and I've never seen a team need to bye more than this team."
In a game containing the first scoreless opening half in five seasons, the Bulldogs were directionless under the guidance of Burton and Galvin.
They finally broke the deadlock with 47 minutes gone through Sitili Tupouniua, then Crichton followed him over the line three minutes later.
Parramatta, to their credit, refused to die wondering and jagged two tries back through Da Silva and Joash Papalli to square the game up with 19 minutes left.
But Da Silva brought all the Eels' hard work undone.
First the hooker gave away an escort penalty on Tupouniua on a goal line drop-out when the Eels had their rivals under mountains of pressure.
Then as Burton went on the charge, Da Silva appeared to make the slightest contact with the head of the Bulldogs five-eighth.
Crichton stepped up to sink the winning goal from 40m out.
"I think he (Da Silva) did make contact with his head, just maybe. So them's the rules," said Eels coach Jason Ryles.
"When you slow it down enough, and the players lay down and wait for the penalty ... it is what it is."
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