The glory went to Aiden Matranga (pictured) after 65 minutes, although Karl Charlton had a goal disallowed and also hit the bar, while Dave Walker had a effort cleared off the line.
Grove manager Peter Ward, said: "It was a good game and I could see why Rainworth had taken 10 points out of the last 15.
"They came flying out of the traps, but once we got a foothold they did not cause us any problems.
"We scored the goal that mattered following a three-man move which ended with Matranga finding the net with a shot across the keeper.
"I made a few changes after last week's 3-0 defeat at Newcastle Town.
"Goalkeeper Steve Hodgson came back, Wayne Johnson played at the back and Liam Shotonton partnered Walker up front with Matt Haddrell on the bench.
"Conditions were poor, but we still tried to get the ball down and play. It was a hard-fought game and we came out on top with a disciplined performance."