Northampton winger Ashton, 24, denied the charge of unsportsmanlike conduct for the incident, which occurred in the first half of Northampton's 30-25 defeat at Welford Road and sparked a touchline brawl.
Tuilagi and Saints flanker Tom Wood were both sent off for punching. A disciplinary hearing on Tuesday deemed their red cards to be sufficient punishment.
Ashton will miss Northampton's back-to-back Heineken Cup fixtures against Castres, the Christmas Eve Aviva Premiership clash with Bath and the New Year's Day trip to Newcastle.
The England wing will be available for Northampton again in time for the Friday night Premiership game against Harlequins on January 6.
RFU disciplinary officer Judge Jeff Blackett, who chaired the three-man panel, said: "We accepted that this started by accident but it became clear that at some stage Chris Ashton knew what he was doing.
"If you pull somebody by the hair there is an element of denigration and this act was a catalyst for a mass brawl that resulted in two red cards and significantly affected the image of the game.
"Without any laid-down guidance on sanction we have aligned the offence to that of spitting and reduced the entry level points to that of three, six and 10 weeks for low, mid-range and top end respectively.
"This was a mid-range offence but we have reduced the sanction by two weeks based on the player's good record and conduct at the hearing."
Ashton, who was also ordered to pay £500 in costs, has 48 hours to lodge an appeal once he has received the full judgement.
Citing commissioner Mike Rafter issued level one citings - the equivalent of a yellow card - to Leicester duo Geordan Murphy and Horacio Agulla and Northampton lock Courtney Lawes for their roles in the mass brawl.
The players were not required to appear before the disciplinary panel and can continue playing.
Earlier in the evening, Northampton forward Phil Dowson was banned for two weeks after pleading guilty to a dangerous tackle on Saracens full-back Alex Goode in the Saints' 30-8 victory on November 26.
Dowson will miss the two Castres games and he was also ordered to pay £500 in costs.