Arsene Wenger's men failed to take full advantage of Chelsea's shock 3-2 loss at Stoke City on Saturday, although they did extend their lead on their London rivals - and Liverpool - to five points at the Emirates Stadium.
Barcelona loanee Deulofeu struck with six minutes remaining as Everton ended an excellent week, which also featured victory over Manchester United at Old Trafford, on a high.
The visitors controlled the first half but lacked cutting edge and, in an end-to-end second period, Arsenal looked to have secured three points when Mesut Ozil converted from close range in the 80th minute.
However, substitute Deulofeu – who was introduced just five minutes before scoring – had other ideas, firing past Arsenal goalkeeper Wojciech Szczesny to rescue a point.
Olivier Giroud almost found a dramatic winner for Arsenal, firing a long-range effort against the crossbar deep into stoppage time.
Giroud and Jack Wilshere returned to Arsenal's starting XI, while ex-Everton midfielder Mikel Arteta and Kieran Gibbs also came in.
Everton were unchanged and carried on where they had left off against United on Wednesday, dominating the early stages with their short, sharp passing.
For all Everton's superiority, they did not muster a clear-cut chance, with Ross Barkley, Sylvain Distin and Kevin Mirallas all failing to hit the target from their side's most promising forays forward.
Kieran Gibbs came agonisingly close to connecting with a teasing Santi Cazorla cross at the other end in the 27th minute.
Arsenal finally awoke from their slumber in the final five minutes of the first half, and Everton goalkeeper Tim Howard was forced to rush off his line to initially save from Giroud before also denying Aaron Ramsey.
The second half saw far more chances on goal, with Cazorla first to go close when he headed straight at Howard after a clever Wilshere flick.
Steven Pienaar - set free by a Barkley backheel - then forced Szczesny into a flying save, before an acrobatic Ramsey volley, that looked to be creeping in, was kept out by Howard.
Barkley shot at Szczesny as the final 20 minutes approached, while Arsenal manager Wenger opted to make a triple substitution, introducing Theo Walcott, Tomas Rosicky and Mathieu Flamini as he went for victory.
Flamini almost had an immediate impact, shooting just wide, before Mirallas and Distin went close for Everton.
Ozil gave Arsenal the lead when he converted a Walcott knock-down from a Rosicky cross, but the hosts were in front for just four minutes as Barkley's run and Bryan Oviedo's cross set up Deulofeu, who produced a powerful finish.
Flamini and Romelu Lukaku had late chances before Giroud went even closer at the death.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.