Ivan Rakitic headed Barca into the lead just before half-time, but Angel Correa slotted home on the hour mark just seconds after Messi limped off to hand Atletico a deserved point.
Earlier, Real Madrid's record-equalling 16-game La Liga win streak came to an end as Villarreal drew 1-1 draw at the Santiago Bernabeu.
Madrid continue to lead Barca by three points at the top of the table with Atletico a further point back in fourth.
"Now we have to look for other solutions. With Leo we are much stronger, but without him we are still a strong team."
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Messi is set to miss Barca's next three games at Sporting Gijon and Celta Vigo in La Liga and Borussia Moenchengladbach in the Champions League, as well as Argentina's World Cup qualifiers against Peru and Paraguay.
The five-time World Player of the Year missed Argentina's last qualifier in Venezuela earlier this month due to a similar injury.
Messi had the first effort on goal, but his fierce effort was comfortably handled by Jan Oblak in the Atletico goal.
The hosts had to wait until four minutes before the break to seriously threaten again when they took the lead.
Atletico momentarily switched off as a corner from the left was played short to Andres Iniesta and his cross was powered home by the late arriving Rakitic. However, Barca's joy was short-lived, first by a double injury blow and then Correa's leveller.
The Argentine's replacement Arda Turan had barely taken up his position when Atletico levelled as Correa coolly slotted home after Javier Mascherano's slip offered him a clear path to goal.
However, in the end both settled for a point which keeps them within touch of Real.
At the Bernabeu, the European champions were looking to surpass the Liga mark of 16 wins first set by Pep Guardiola's Barcelona in 2010/11, but were made to pay for a sluggish first-half display.
"The second-half was great from everyone, but you can't always win the three points at the end.
"We know we have to work hard and change things, above all how we start the game."
Cristiano Ronaldo and Gareth Bale returned to the Madrid side that won 2-0 at Espanyol on Sunday.
However, Villarreal were the better side in the first-half and got their reward just before half-time when Sergio Ramos handled Manu Trigueros's shot and referee Jose Luis Gonzalez pointed to the spot. Captain Bruno Soriano stepped up and coolly chipped the ball down the middle of the goal past Casilla.
Ronaldo was then inches away from turning home Dani Carvajal's low cross and Bale curled just wide as the pressure increased from Madrid.
However, their frustration spilled over when Ronaldo earned a yellow card for kicking out at Bruno. The off-colour Bale was then replaced by Lucas Vazquez, whilst Karim Benzema made way for Alvaro Morata in the search for a winner.
Morata's header from Vazquez's cross was brilliantly turned over by Asenjo and when Ronaldo fired straight at the Villarreal stopper in the final minute, Real's shot at history was gone.