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The 23-year-old Exeter prop committed the offence in Wales' 25-21 defeat on Saturday.
He will now miss the rest of his club's regular season games, although he would be eligible to play in the European Champions Cup final if Exeter make it.
Francis can return on Monday, 9 May.
Wales assistant coach Rob Howley says Francis was unaware of what he was doing in the incident that led to him being disciplined.
"When you look at it, his [Francis'] eyes are closed and I don't think he knows what he's doing," said Howley.
"He went to clear someone out."
The second-half incident was reviewed by the television match official and resulted in a penalty against Wales.
World Rugby's recommended sanctions for those found guilty of such offences range from between 12 and 208 weeks but the body said it viewed the offence "as being at the lower end of the scale".
"The disciplinary committee was then required to consider aggravating and mitigating factors, which it did before concluding that the appropriate sanction be a playing suspension of eight weeks," World Rugby added.
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Wales coach Warren Gatland was unsure in his initial assessment of the incident.
"I haven't seen it again, and I listened to the TMO comments on it and he said there was only one view of it and it was inconclusive," said Gatland after the game.
"He thought there was contact with the face.
"I'll be honest and say my initial reaction to it was it didn't look great, but I haven't had a look at it since then."
Francis was a second-half replacement for Samson Lee at Twickenham.
Wales prop Tomas Francis has been banned for eight weeks after appearing to make contact with the "eye or eye area" of England's Dan Cole and will miss the Six Nations match with Italy.