Simon (9-3) allowed just one run on seven hits, striking out five without a walk, over a career-high eight innings for the Reds, who won the final two games to earn a series split.
The right-hander has won his last three starts, surrendering just seven runs and one walk in 20 1/3 innings of work over that span.
Todd Frazier added a two-run homer for Cincinnati, which has won four of its last six to salvage a 5-5 mark during a 10-game homestand.
Greinke (8-3) had a rough first inning and his offense couldn't bail him out, as the Dodgers ended a seven-game road trip at 4-3. The right-hander was charged with just two runs on only four hits over six innings.
The Reds wasted little time responding and got to Greinke before he could even get warm. Skip Schumaker started it off with a single to center and Frazier followed with a homer.
Joey Votto followed with a double, but Greinke got out of any further trouble by getting Brandon Phillips and Jay Bruce to ground out and struck out Devin Mesoraco to end the frame. Phillips was initially ruled safe, but a replay review overturned that call.
Greinke continued to dial it up over the rest of his outing, allowing just one hit, a Votto single with two outs in the third, the rest of the way.
Simon also settled down after a hiccup in the first. He allowed just one hit and hit one batter over the next five innings and then got out of a two-on, no-out jam in the seventh to preserve the lead.
With Kemp on third and Andre Either at first after back-to-back singles to start the frame, Simon got Tim Federowicz to fly out to shallow right to hold the runners. Then, Miguel Rojas' grounder to third ended in Kemp being tagged out after a rundown for the second out, and pinch-hitter Scott Van Slyke struck out to end the inning.
The Reds added two insurance runs in the eighth. Zack Cozart started the inning by greeting reliever Brandon League with a double and later scored from third on a wild pitch. Schumaker also doubled in the frame and came around on a Ryan Ludwick pinch-hit single to cap the scoring.
The Dodgers threatened against closer Aroldis Chapman in the ninth, putting runners on the corners with one out. However, Chapman struck out Rojas and pinch-hitter Jamie Romak to end the game and earn his 10th save of the season.