Lambert then doubled the lead again from the spot ten minutes into the second half.
The 29-year-old was brought down by Adrian Mariappa and stepped up to take his tally for the season to eight.
Guly do Prado made it three, securing the three points in the 70th minute.
It was four when Lee Holmes cut in from the right before curling a shot into the far corner.
After the upcoming international break, Southampton travel to Derby while Watford, who drop to 17th, host Crystal Palace.