Striker Teemu Pukki scored from a counter-attack in the 79th minute, his team's first attempt on target after they had withstood waves of attacks from Spain.
Defender Sergio Ramos, wearing the captain's armband on his 100th appearance, had broken the deadlock with a header from a corner just after half-time but Vicente del Bosque's side struggled to make their dominance count.
The draw dropped Spain down to second place and they are now under pressure to win in Paris on Tuesday when they visit new leaders France who overcame Georgia 3-1.
Les Bleus have 10 points from four games while Spain are on eight, having drawn 1-1 with France in October.
"We weren't thinking about France because this was the most important game," said the 26-year-old Ramos who became the youngest Spaniard to rack up a century of internationals.
"Now we can think about it," he told Spanish television. "We have a great team in front of us but we can beat them and we have to go with this mentality.
"This result is a shame after celebrating my 100th cap with a goal," added Ramos.
It was an unfamiliar Spain line-up without injured stalwarts Iker Casillas and Carles Puyol or rested midfielders Xavi and Xabi Alonso but the script was unchanged as the hosts had more than 80 percent possession of the ball.
The Finns defended with 10 men in their area at times, restricting the opposition to shots from distance as Spain showed a lack of ideas when it came to picking their way through the massed white ranks.
Coach Del Bosque, who passed Ladislao Kubala's record with his 69th match in charge, threw on Pedro for Santi Cazorla at the interval in a bid to pep up the attack and it was Ramos who made the breakthrough.
The visitors looked like they might fold under the pressure but they stood firm, slowly grew in confidence and as Spain appeared to ease off the Finns broke away to equalise.
In a desperate closing few minutes substitutes Alvaro Negredo and Juan Mata went close for the home team and Finland nearly grabbed a second on the counter-attack.
Tempers almost boiled over following a goalmouth scramble when David Silva picked up a booking that rules him out of the game in Paris.
"We failed to get the shots in to shut out the game with a second goal," said Del Bosque. "They defended from deep very well and what seemed almost impossible happened - they scored.
"We still hold our destiny in our own hands. We have a great team and it is possible to win in France."