Barca's all-time leading scorer Messi brought up the milestone at Camp Nou with a brilliant Panenka from the penalty spot as he punished Felipe for sending Nelson Semedo to ground early in the second half.
However, Saul, who first equalised in the opening period after Diego Costa's own goal, powered home from 12 yards for the second time after Semedo caught Yannick Carrasco in the box.
The result provided a boost to Real Madrid, who can open up a four-point advantage at the summit by beating Getafe on Thursday.
Messi appeared to have ended his wait for his landmark goal in the 11th minute, but his corner only beat Jan Oblak due to a deflection off the inside of Costa's leg.
Marc-Andre ter Stegen saved a penalty from Costa after Arturo Vidal tripped Carrasco, only for a VAR review to spot the goalkeeper had strayed off his line. A re-take was ordered and Saul guided the ball into the bottom-right corner.
Messi curled a magnificent attempt from the right corner of the box narrowly wide, before steering a cutback from Jordi Alba outside the left upright and seeing a deflected free-kick turned behind by Oblak before half-time.
Barca regained the lead five minutes after the restart when Messi cheekily chipped home his 700th goal from the penalty spot.
Costa missed the target with a header but Carrasco won another penalty and Saul's spot-kick found the net off the gloves of Ter Stegen.
Vidal was unable to find the target with a pair of decent opportunities as Barca were forced to settle for a draw that will do little to quell the rumours of discontent in the dressing room.
What does it mean? Barca falter again
Quique Setien changed his approach for the visit of Atleti, leaving Antoine Griezmann and Ansu Fati on the bench and playing Riqui Puig behind a front two of Messi and Luis Suarez.
The Blaugrana dominated possession throughout but proved vulnerable on the counter-attack and Setien did not change formation until he introduced Fati in the 85th minute – Griezmann came on five minutes later.
It was too late and Barca have now drawn three of their past four LaLiga fixtures and seen the two-point lead they held when the season was interrupted by the coronavirus pandemic evaporate. With five games remaining, Setien needs to turn things around quickly.
Magical Messi not enough
Everything positive for Barca was funnelled through Messi – he had more shots in the first half than the entire Atleti team. After three games without a goal he reached his milestone with class, but Diego Simeone's men were still able to snatch a point.
Costa's miserable outing
After narrowly missing out on connecting with a teasing Carrasco free-kick in the sixth minute, Costa put the ball in the net at the other end. His penalty was poor and he was fortunate VAR offered Atletico a second chance to equalise.
Atleti are back in action at home to Real Mallorca on Friday, while Barcelona face a tricky trip to Villarreal on Sunday.