Dallaglio, who faces a Rugby Football Union disciplinary hearing in London next Tuesday as a result, will now miss Wasps' Guinness Premiership appointment with Leicester this weekend.
The former England captain's club ban took effect two days ago.
Wasps say they are satisfied there was no intention or malice on Dallaglio's part, although his actions did infringe the relevant International Rugby Board rule.
The club also took into account what they describe as Dallaglio's "exemplary previous disciplinary history."
Dallaglio said: "I accept the sanction of a week's suspension from playing for my club.
"I acknowledge that my actions infringed the rules, although I wish to stress that they were unintentional and without any malice.
"I have extended my apologies to Julian White.
"I hope that I can now draw a line under this incident and enjoy playing out the rest of my final season."
Dallaglio, 35, will retire from all rugby next month following a glittering career that saw him help England win the 2003 World Cup.
He won 85 caps - only Jason Leonard is above him on England's all-time list - in addition to touring three times with the Lions.
Wasps are currently battling to reach the Guinness Premiership play-off positions in their quest for a fourth league title in six seasons.
They have seven games left, and a high Premiership finish is now their solitary route into next season's Heineken Cup after making an early European exit this term and bowing out of EDF contention against Leicester.