England international Hartley caught Tigers centre Smith off the ball during Saturday's 23-19 Premiership win.
Hartley, 28 pleaded guilty and will miss games against Harlequins, Newcastle and Sale. It is the fifth time he has been banned.
England start their Six Nations campaign against Wales on 6 February.
An elbowing offence carries a minimum two-week ban, but judicial officer Jeremy Summers said the suspension "reflects the player's record rather than the seriousness of the offence itself".
In April 2007, Hartley was banned for 26 weeks for gouging Wasps forwards James Haskell and Jonny O'Connor.
He was suspended for eight weeks in March 2012 for biting Ireland's Stephen Ferris while playing for England in the Six Nations.
And he was banned for two weeks later that year for striking Ulster's Rory Best.
Verbal abuse of a Premiership match official in 2013 earned him an 11-week ban, costing him a place on the British and Irish Lions tour of Australia.
Hartley will be available again for Northampton's European Cup game against Ospreys at the Liberty Stadium on 18 January.
Hartley, who was born in New Zealand, has won 61 caps for England since making his debut in 2008.
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