Just been looking back through Soccerbase, and we scored 5 (or more) in consecutive home league games in the 1983/1984 season.
28th April 1984: Saints 8-2 Coventry
7th May 1984: Saints 5-0 Tottenham
However, there was an away game sandwiched between them, a 1-1 draw at Stoke. Will look further back for actual consecutive games tomorrow, unless someone gets there before me