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However, Kahn Fotuali'i cut the gap for Northampton just before the break with his first ever Premiership try, leaving the home side five points down at half time.
Williams was able to capitalise on a number of Saints errors in the second half, adding three kicks in the space of seven minutes to move the visitors into a comfortable lead.
Ethan Waller managed to cut the gap late on, but Northampton ran out of time as they looked to level the score in the final five minutes.
The result leaves Leicester seven points behind the second-placed side in the table.