The Wrens dominated the home side for the majority of the game, putting in a much-improved performance following defeat to Belper United the previous Saturday.
The first half started lively with the visitors having the better of possession.
Rainworth came close to taking an early lead in the third minute as Isaac Fisher's curling shot dropped narrowly wide of the left post having worked in well from the right wing.
The Wren's would make the most of their early dominance as Kane Hempshall caught Tevvy keeper Louis Pryce off guard. Some smart defensive play from skipper Ryan Herbert allowed Rainworth to break down field before Hempshall fired the ball from the edge of the box into the bottom right corner.
It would take just seven minutes for Rainworth to double their lead. An in-swinging cross from Joe Naylor found Julian Topliss who headed home with ease as the Teversal defence stood off.
The home side were unfortunate not to get one back in the 28th minute as a long-range Tom Batterham freekick whistled wide of the top right corner.
Rainworth almost found a third just two minutes before the break as Clarke teed up Andy White on the edge of the box as his shot kissed the crossbar on its way over.
The away side did find their third three minutes into the second half. An in-swinging Isaac Fisher corner was headed home by Kyle Hempshall as he broke free of his marker inside the box.
Teversal managed to get a goal back in the 74th minute as Ben Agbakosi latched onto a chipped through ball before firing home well inside the box.
However, the Tevvy revival would be quashed by a moment of high quality from Wren's keeper Warren Squires. Winger Henry Brooks was found superbly on the left by Squire's long kick allowing Brooks to dance inside before arrowing his shot into the bottom right corner.
Brook's goal wrapped up a needed result from Rainworth concluding their highly efficient performance at Tomlin Personnel Park. Next up, Rainworth face an away trip to Graham Street Prims, a fixture they won 4-1 last season in the final game before lock down.
Teversal: Pryce, Shorthouse, Bower, Hudson, Smith, Geeson, Widdowson, Atkins, Batterham, Simpson, Smith
Subs: Agbakosi, Bacon, Brentnall, Summers, Buttery
Rainworth: Squires, Herbert, Renshaw, Jones, Kyle Hempshall, Naylor, Clarke, Kane Hempshall, Topliss, White, Fisher
Subs: Harris, Darby, Brooks, Coupe, Wootton