But Nuno Espirito Santo's men ensured they would avoid defeat in their first Premier League match since the 2011-12 season when Jimenez – one of four new faces in Wolves' starting XI – nodded home Ruben Neves' cross.
Neves had earlier levelled proceedings with a classy free-kick straight after Jagielka's red, and Marco Silva will reflect on a frustrating debut outing as Everton boss.
A rocking Molineux was silenced in the 17th minute. Michael Keane attacked a Leighton Baines free-kick that Wolves failed to deal with and Richarlison prodded home from close range.
The pyrotechnics that preceded kick-off were not initially followed with fireworks by the hosts on the pitch, but the complexion of the match changed four minutes before the break when Jagielka lost control of a routine pass then inevitably saw red for an ugly lunge on Diogo Jota.
From the resulting free-kick Neves curled an unstoppable effort into the top-right corner.
Cenk Tosun and Jimenez exchanged missed efforts before Everton regained the initiative after 67 minutes.
Richarlison combined with Tosun on the left of the area and coolly stroked home into the far corner.
But Wolves fought back again and Jimenez earned a share of the spoils 10 minutes from time when he met Neves' excellent left-wing cross with a glancing header past Jordan Pickford.
What it means: Wolves up for the fight
Much is expected of ambitious Wolves this season and, on this evidence, they remain a work in progress.
But what is not in question is the fight in the ranks, while the class of Neves and Joao Moutinho coupled with the industry of Jimenez, Diogo Jota and Helder Costa means they will cause plenty of teams problems.
It was not quite the start Silva was looking for, but Everton have the added bonus of several deadline-day signings, including Yerry Mina, Andre Gomes and Bernard, to come into contention.
Pat on the back: Marco to be Richarlison’s Silva lining?
While many critics questioned the wisdom of paying such a big fee for a player without a Premier League goal since November, there is no doubting Richarlison's best football at Watford last season came under Silva.
A reunion under the Portuguese may be just what Richarlison needs to rediscover the form he showed in his early days in England's top flight.
Boot up the backside: Jagielka red paves way for Mina, Zouma
Even the most casual of football followers will know that things can change in the blink of an eye, as proved by Jagielka's moment to forget.
The Toffees were in complete control before his error of judgement and rash challenge on Jota meant Everton's grasp weakened.
With Mina and Kurt Zouma soon to be available it may end up being a very costly red for Jagielka.
Key Opta stats:
7 - Ruben Neves has scored seven league goals from outside the box since the start of the 2017-18 season, more than any other player in the top four tiers of English football. Range. pic.twitter.com/vOE6Nk79Ii— OptaJoe (@OptaJoe) August 11, 2018
- Richarlison found the net with his first two shots in the Premier League for Everton, having failed to score with any of his last 53 shots as a Watford player.
- Everton defender Phil Jagielka was sent off for only the second time in his Premier League career – he is the oldest Englishman (35y 359d) to be red carded in the top-flight since Chris Powell (37y 81d) in November 2006.
- Everton’s red card was their 90th in the Premier League – more than any other side.
- Wolves defender Matt Doherty made his second Premier League appearance – 2,513 days after his first in September 2011; only two outfield players have had a longer wait for their second appearances – Erik Nevland (3,667) and Darren Pratley (2,835).
- Ruben Neves both scored and assisted for Wolves – the last Portuguese player to achieve that in a Premier League game was Nani in April 2012, also against Everton.
Everton will be frustrated at taking a point from a game they largely controlled and will look to get their first win of the season at home to Southampton, while Wolves' first away trip of the campaign comes at Leicester City.