Richarlison added a fine second after the break, but Raul Jimenez headed home from six yards on his debut to give the Championship winners a point on their return to the top flight.
Both teams went into the game with their respective supporters expecting glimpses of exciting football, but the opening stages in the West Midlands were littered with wayward passes and no significant openings.
However, it did not take the Toffees much longer to go ahead. A free kick into the middle was not dealt with by Boly, and the ball fell to Richarlison to fire low into the net from little more than six yards out.
Conceding so early after their return to the top flight took the sting out of Wolves, and Nuno Espirito Santo could only watch on as his side was restricted to speculative long-range strikes.
Everton looked in control of the game with five minutes of the first half remaining, but a mistake from Jagielka gave Wolves a way back into a match in which they were being outclassed.
Jagielka should have controlled a routine pass across the edge of the area but, after the ball got away from him, the former England international slid in late on Diogo Jota with his studs and gave the referee reason to dismiss him for dangerous play.
Neves was always going to take the resulting free kick from 20 yards, and the Portugal international managed to get the ball over the wall and into the top corner of Jordan Pickford's net.
The onus was on Wolves to push men forward during the second half, but the home side continued to show caution and it only played into the hands of Everton.
Cenk Tosun should have done better when presented with a two-on-one scenario in a central position, but the Turkish frontman saw a shot from 20 yards tipped behind by Rui Patricio when he should have played in Theo Walcott.
Wolves responded by going on the front foot and a through-ball from Ryan Bennett found the run of Jimenez, but the Mexican opted for power instead of placement and Pickford made an excellent block.
Everton continued to match Wolves blow for blow, however, and midway through the second half, the visitors regained their advantage when Richarlison was given the freedom to run towards goal and curl a low effort into the far corner of Patricio's net.
Wolves' equaliser arrived from the head of Jimenez, who was perfectly placed at the back post to meet Neves's inswinging cross and power an effort inside the near post from six yards.
Both sides refused to settle for a point during the closing stages, but a share of the spoils in the Wolverhampton rain was a fair result.
WOLVERHAMPTON WANDERERS (3-4-3): Patricio; Bennett, Coady, Boly; Doherty, Neves, Moutinho (Gibbs-White 85'), Jonny (Vinagre 76'); Costa (Bonatini 71'), Jota; Jimenez
EVERTON (4-3-3): Pickford; Coleman, Keane, Jagielka, Baines; Gueye, Schneiderlin; Walcott, Sigurdsson (Holgate 43'), Richarlison (Digne 86'); Tosun (Niasse 81')body check tags ::