Lynch scored on runs of 15 and 31 yards among his 28 carries for the top-seeded Seahawks, who advanced to the NFC Championship Game for the second time in franchise history.
They will next host the winner of Sunday's other NFC divisional round contest between San Francisco and Carolina.
Russell Wilson completed 9-of-18 passes for 103 yards and ran three times for 16 yards for Seattle, which won despite picking up just 277 yards of total offense in rainy conditions.
Colston recovered the ensuing onside kick but the Saints couldn't advance, ending their first playoff run since 2010.
Colston finished with 11 catches for 144 yards and a TD.
Khiry Robinson rushed for 57 yards with a score on 13 carries for New Orleans.