Signed for £12m from Marseille in the summer, Thauvin was part of a strong Newcastle side named by manager Steve McClaren as he showed his intent to make an impact in the cup competitions this season.
His goal on 3 minutes and assist for Siem de Jong on 8 minutes raised fears that the visitors could be in for a hammering but Marc Richards halved the deficit from the penalty spot almost immediately.
The scoreline stayed the same until 10 minutes into the second half when another quickfire double from the hosts secured their passage into the next round.
Northampton's attention turns to the weekend and the return to League Two action while their share of a near-27,000 gate will also be useful should Chris Wilder look to make more additions in the transfer market.