Looking to keep their title hopes alive, Louth lost the toss and were put in on a damp wicket.
Openers Dean Wright and Matt Leeming started steadily, but once the first wicket fell With the score at 24, the pitch came to life, helping reduce Louth to 57-5 after 24 overs.
Wright knew staying at the crease was important to give his bowlers a chance and batted with caution and skill to make a defiant 62 not out, including five boundaries.
Supported by Tom Ryder (32) Louth posted 132 in their 40 overs, a reasonable total considering the difficult pitch and slow outfield.
Louth knew early wickets were key to pressure Southbank, and duly took them with Luke Johnson (6-0-15-2) and Keaton Critchell reducing Southbank to 14-3.
Home batsman Laistor batted with discipline to keep their victory hopes intact, but he lacked support as regular wickets fell, with Keaton taking three scalps.
With the score at 103-7, Louth skipper Zain Siddique (1-1-0-2) came into the attack and bowled a match-winning over of leg spin, a double-wicket maiden to all but complete a memorable win.
Critchell bowled their last batsman to finish with impressive figures of 8-2-19-4 as the hosts were all out for 103, handing Tavs victory by 31 runs.
Louth host Brocklesby Park on Sunday.