Sola found Alvaro Cejudo's pass down the right flank in the fifth minute and netted past goalkeeper David Cobeno with a chip shot from a tight angle.
Four minutes later Rayo nearly equalised but Osasuna keeper Andres Fernandez impressively blocked a shot from Francisco Medina.
Osasuna remained unbeaten in its last four games, while Rayo stayed in the middle of the standings.
Kike Sola was on the pitch for 63 minutes against Rayo Vallecano. The striker scored in the fifth minute.
Los Rojillos have now recorded a record of three wins, seven losses and four draws this season, which puts them in 16th place, only one point above the relegation zone.
With 19 points Los Franjirojos trail Levante by one for the final Europa League spot, and have compiled a record of six wins, seven losses and one draws on the season.
En route to the win, Osasuna had the ball for 38.5% of the match. Osasuna had 12 shots, including three on target.
Joseba Llorente made a big difference on the match for Rojillos after coming on for Sisi in the 35th minute due to injury, as he took two shots.With a ball control percentage of 61.5%, Rayo Vallecano played an effective brand of possession football despite the loss.
The nine shots that the Rayistas managed were not enough to secure the win. They also completed 279 passes and had 36 crosses in the loss.
Piti had three shots for Rayo Vallecano, more than any of his team-mates.
The combination of Nacho to Roberto Trashorras was the most common during the match, as the two players connected 13 times.