Mekies, who took over at Red Bull following the dismissal of Christian Horner in July, said his team had gone too far in interfering with the tape used to indicate Lando Norris's position on the grid.
A Red Bull team member entered the grid area after the start of the formation lap at last Sunday's United States Grand Prix and tried to tear the tape off the pit wall.
"I think there has been a bit of silly games being played for a few races between the teams, and it probably got out of hand," Mekies told Sky Sports. "Moving forward, it will be more in control."
Norris, who finished the race second behind four-time champion Max Verstappen, said he thought it was 'amusing'.
"Good job by them because they can, but it didn't matter," he said. "I didn't use the tape. So, it was extra amusing because I didn't need it.
"We just put it there in case, so it made it extra funny because they got a penalty for it, and I didn't even need it."
Gamesmanship to gain an advantage by any means has been common in Formula 1 for many years, but it is rarely uncovered and punished.