Seven-times world champion Schumacher admitted liability for the accident that occurred just after the midway point of the race at the Marina Bay Street Circuit.
It followed the restart after a safety car period for HRT's Narain Karthikeyan hitting a barrier across Anderson Bridge.
In what was a repeat of two previous incidents - one at this track last year and one earlier this season when Schumacher ran into the rear of Williams' Bruno Senna in Barcelona - the stewards decided to heavily penalise the 43-year-old for the next race in Japan.
Schumacher collected a five-place penalty for the subsequent race in Monaco after hitting Senna, one which saw him start sixth after he had set the fastest time in qualifying.
On this occasion, given it was a repeat offence, the punishment was doubled, with the stewards stating: "The penalty takes into account this is the second similar offence by the driver this season.
"The driver admitted the collision was his error due to the failure to anticipate the braking performance of the car with lower grip following a safety car period."
For his part, Schumacher said: "It was obviously a very unfortunate ending to my race when I ran into the car of Vergne, who accepted my apology straight afterwards.
"I am not totally sure why it happened. I was braking, but the deceleration was not as strong as it usually would be and I could not avoid running into the car in front of me.
"We have to find out what happened. Up until then I think it would have been possible to get some points."
As the two men climbed out of their cars, Mercedes driver Schumacher waited for Vergne to come across, immediately putting his arm around the Frenchman before they headed back to their respective garages.
Toro Rosso driver Vergne said: "I was focussing on catching Perez at that point, trying to brake late to catch him, so I am not too sure what happened exactly.
"I assume Michael braked a bit too late and could not avoid running into me.
"There is no sense in being angry about it because these things happen in racing and even the most experienced driver on the grid can make mistakes.
"He said sorry and that's the end of it."