Swans keep the pace in UCL title race
Newport trailed to Jordan Pendered's sixth minute strike before Lawless reopened his account on 28 minutes to level things up for the Swans.
The scores remained level into the second half, but Michael Lyon's quick-fire brace, scoring twice in three minutes, put the home side firmly in control at 3-1.
Advertisement
Hide AdAdvertisement
Hide AdAnd when Lawless added his second - a wonderful strike with nine minutes to go - Newport had the three points wrapped up.
The result sees Darren Lynch's side remain third, a point behind Wisbech Town, while leaders Yaxley are seven points clear having played three games more than the chasing pair.
Newport take on 14th place Rothwell Corinthians this Saturday.