Warning: Some people may find the below images graphic.
The former England rugby star uploaded a series of nine images of his son showing the devastating effects of this illness.
He praised Great Ormond Street hospital for treating his toddler, Sami, and encouraged others to "vaccinate now".
"The two weeks of hell we've just had cos of meningitis," Dawson tweeted. "Sami lucky due to amazing people @GreatOrmondSt."
Dawson said he and his wife had deliberated over whether or not to share the heartbreaking images.
"Been umming and ahhhing about whether I should write the next post but my wife and I have agreed it's important. The pic may be upsetting," he wrote.
Dawson, who is also dad to Alex, four, with wife Carolin Hauskeller, was overwhelmed with the support he received after posting the images.
He encouraged people to sign a petition, set up by a mother whose daughter died of meningitis on Valentine's Day, which calls for all children to be vaccinated, not just newborn babies.
"We're overwhelmed by all the messages for Sami. I ask one favour. Please read and sign so fewer kids suffer, thanks."
Dawson later thanked the staff at Chelsea and Westminster Hospital for recognising his son's disease.
"I must also heap praise on @ChelwestFT for their superb A&E and recognition of Sami's disease. We're indebted... #lifesavers #vaccinateNOW."
Symptoms can appear in any order. Some may not appear at all.
For babies and toddlers:
Fever, cold hands and feet, vomiting, refusing food, dislike of being handled, sleepy, drowsy, floppy, unresponsive, pale/blotchy skin, spots/rash, crying, soft spot on child tense, child disliking the light, stiff neck and convulsions.
Fever, cold hands and feet, vomiting, drowsy, difficult to wake, confusion and irritability, severe muscle pain, pale and blotchy skin, spots/rash, severe headache, stiff neck, dislike bright lights and convulsions/seizures.
If you spot these symptoms, the NHS advises you should "seek medical help immediately", stating: "This may mean going to the accident and emergency (A&E) department of your local hospital in the middle of the night".
For more information on meningitis symptoms and signs to look out for, visit MeningitisNow.org or call their helpline number 0808 80 10 388.