The hosts lined up without top scorer Dominic Calvert-Lewin, who is battling a groin problem. Tottenham Hotspur deployed a three-man defense, with Toby Alderweireld returning to the starting lineup at the expense of Lucas Moura.
A cagey start to the game saw both sides struggle to find their rhythm. However, it was Everton who had the first real chance when Richarlison was played through on goal. The Brazil international, however, shot straight at Hugo Lloris when he should arguably have done better.
The visitors made them pay soon after when Harry Kane showed great composure to control the ball before unleashing a pristine left-footed finish beyond the dive of Jordan Pickford.
Tottenham Hotspur's lead lasted just four minutes as Sergio Reguilon clumsily bundled into James Rodriguez, and Sigurdsson converted the ensuing spot-kick to restore parity on the half-hour mark.
There was still time in the first half for Lloris to pull off a stunning save from Rodriguez and ensure that both sides went into the break all square.
Tottenham Hotspur started the second half on the stronger foot and went close on two occasions through Heung-Min Son and Toby Alderweireld. Richarlison found the back of the net on the hour mark only for the linesman's flag to pull him up for offside.
There was no denying Everton just two minutes later, though, as a brilliant counter-attacking move saw Seamus Coleman send a delightful ball into the box, which Sigurdsson finished with an emphatic left-footed volley.
Interestingly, Coleman had been on the field for less than a minute, having replaced Alex Iwobi in the 61st minute.
The goal prompted Jose Mourinho to make changes of his own, with Erik Lamela and Lucas Moura both entering the fray.
The ping-pong nature of the game continued as Everton's lead lasted for just six minutes. Lamela's cross into the box was not properly dealt with, and Kane was on hand to draw Tottenham Hotspur level.
The goal took him to 164 goals in the Premier League and 7th on the all-time standings.
Kane hit the outside of the post soon after, and Richarlison had a perfect chance to win the game for Everton late on.
A pristine through-ball by James found Joshua King in the area, but Lloris pulled off a fine save to deny the Everton forward. The rebound fell to Richarlison on the six-yard line. However, the 23-year-old inexplicably blazed over.
On the overall balance of play, a draw was probably the right result, although it was far from the outcome that either side needed.
Here are five talking points from the game.
#5 Have Tottenham Hotspur's top-four hopes been extinguished?
It seems like eons ago, but as recently as November, Tottenham Hotspur were seemingly in the race for a maiden Premier League title.
However, a spectacular implosion since then has left the Lily Whites scraping for a top-four finish.
The north London outfit came into this game knowing that anything other than a win could put paid to their top-four hopes. Having gone ahead through Kane's well-taken opener, it seemed like they were on their way to three points.
However, Tottenham Hotspur have been characterized by a lack of spine this season, evidenced by the fact that only Brighton and Hove Albion have dropped more points from winning positions than them.
This is quite unlike any Mourinho teams in the past, and this draw made it three league games in a row when Spurs dropped points from winning positions.
Where it leaves them on the table is even more important. With just six games to go, Tottenham Hotspur find themselves four points behind the top four, having played a game more.
A win here would have seen them close the gap on Chelsea and West Ham. However, a failure to get the job done means that the White Hart Lane outfit have all but extinguished their hopes of making the top four.
#4 Impressive Gylfi Sigurdsson steps up to the plate
On his day, Gylfi Sigurdsson is unarguably one of the best attacking midfielders in the league.
However, the 31-year-old has been plagued by serious inconsistency. This has seen him drift in and out of Everton's starting lineup. Fortunately, injuries to midfield regulars saw him handed a start against his former side.
The Iceland international shone like a million stars against Tottenham Hotspur, with his brace helping Everton maintain their quest for European qualification.
He showed great composure to nervelessly convert from the spot in the first half, while the technique required for the second goal underlined his immense ability.
With the Toffees facing a testing end to the season, fans of the club will be hoping that Sigurdsson's performance against Tottenham Hotspur could reinvigorate him to lead their quest for European qualification.