The Toffees, who spent £140m on new players in the summer, have won just two of their opening eight league games this season.
While the draw only lifts Everton to 16th, it might be enough to persuade the club to give Koeman more time.
Koeman praised Rooney for keeping his composure.
"To be fair, the medical staff are overseeing all of it and they're in contact with Everton and all their scientists".
Everton looked marginally more threatening and did eventually have the match's first shot 20 minutes in, but Mat Ryan made a strong save down to his right after Vlasic teed up Idrissa Gueye on the edge of the box.
A corner was cleared as far as Knockaert on the edge of the box and the Frenchman's subsequent volley struck a defender before landing kindly for Dunk, whose effort was crucially blocked behind by Michael Keane.
"If you don't put chances away then what happened can happen".
But Bruno's clumsy elbow into the neck of Calvert-Lewin in the penalty area earned Everton a reprieve as Rooney stepped up to roll in the spot-kick.
The visitors nearly snatched a victor at the death but Australia goalkeeper Mat Ryan pulled off a fine double save to deny Kevin Mirallas.
- Wayne Rooney has now scored against 36 of the 39 different clubs he's faced in the Premier League - only Frank Lampard (39), Andrew Cole (38) and Alan Shearer (37) have scored against more in the competition (Jermain Defoe also 36).
Derby County and Blackpool also thwarted Rooney in two games apiece, while Newcastle United are apparently his favourite opponents.
Brighton is above Everton in 14th thanks to a superior goal difference. We learned that the first game of the season and that was again the case.
McCarthy will not be in the squad for tomorrow's Premier League game against Brighton but will play in an under-23 game against Wolfsburg on Tuesday and could be included against Arsenal next weekend.