Nigel Pearson's team are back in the English top-flight for the first time since 2004 and their return looked set to end in disappointment at the King Power Stadium.
Aiden McGeady put Everton ahead in the first half and, although Leicester's £8 million club record signing Leonardo Ulloa equalised soon after, Steven Naismith gave Roberto Martinez's side the lead again just before the break.
The Foxes became accustomed to finishing strongly as they amassed over 100 points en route to winning the Championship last season and they kept probing at the Everton defence before substitute Wood, who almost joined Championship club Wolves in midweek, snatched the equaliser with four minutes left.
England manager Roy Hodgson was watching from the stands, but he couldn't check on Everton midfielder Ross Barkley who was absent after suffering a medial knee ligament injury in training which is expected to keep him out for up to eight weeks.
Romelu Lukaku and Gareth Barry were available to make their first starts as permanent Everton players following successful loan spells last term.
Former Celtic winger McGeady worked space for a shot on the edge of the penalty area in the early stages, but his effort was straight at Leicester goalkeeper Kasper Schmeichel.
McGeady wasn't to be denied when another chance fell his way in the 20th minute.
A Leighton Baines corner ran through to Sylvain Distin and when Schmeichel raced out to block the defender's shot, the ball fell for McGeady and he brilliantly curled his shot over two Leicester players on the goalline and into the far corner.
Distin was at fault when Pearson's side equalised two minutes later.
His attempted clearance from a corner smashed into Ulloa and the former Brighton forward made him pay by smashing a clinical finish past Tim Howard from close-range.
Lukaku was mostly quiet following his £28 million move from Chelsea and even when the Belgian striker did break through he dragged his cross-shot across the face of goal.
Naismith showed him how to do it as he put Everton back ahead on the stroke of half-time.
A swift exchange of passes between Baines and Steven Pienaar unhinged the Leicester rearguard, allowing Scottish forward Naismith to drive a fine first-time shot in off the crossbar.
Jeffrey Schlupp came off the bench for Leicester in the 63rd minute and the winger had a golden opportunity to equalise with his first touch.
Put clean through on goal, Schlupp completely lost his nerve and blasted high over the bar from the edge of the area.
Barry, already on a booking, was lucky to escape a second yellow card after a clear foul as Leicester launched a counter-attack.
But Leicester had no need to rue that decision as Wood lashed in the equaliser in the 86th minute after Phil Jagielka could only divert a tackle into his path.