Atleti have now not won in their last 15 league meetings with Barca - a run that stretches back to February 2010, almost two years prior to Simeone's appointment.
It appeared as though that miserable streak would finally end on Saturday, as Saul Niguez's stylish first-half strike looked to have set up a victory for the hosts over a Barca team that had won its first seven matches this term.
But leaders Barca built momentum in the second half, with Lionel Messi striking the post and spurning further chances before the leveller deservedly arrived.
Suarez, marginalised in the first half but much improved after the break, stooped to meet Sergi Roberto's cross from the right and headed down past Jan Oblak with eight minutes remaining.
The visitors pushed for a winner, but Atleti held firm to ensure Ernesto Valverde's men failed to win in LaLiga for the first time this season. Their advantage over Real Madrid stands at five points.
Messi had carried Argentina to the 2018 World Cup in midweek with a pivotal hat-trick in qualification and showed no signs of weariness as he wriggled through the Atletico defence inside the opening minute and prodded just wide.
A lively start to the contest continued as Antoine Griezmann tested Marc-Andre ter Stegen at his near post and the same player then skipped beyond Gerard Pique's challenge to draw another smart stop from the goalkeeper.
Pique's involvement was met with jeers as pro-Spanish chants echoed around the stadium - a result of Catalonia's independence referendum, supported by the defender, which was declared illegal in Spain's constitutional court - and his discomfort only increased as Barcelona fell behind after 21 minutes.
Saul took possession at the end of a fine team move to arrow a 20-yard strike across Ter Stegen and into the bottom-right corner.
Although the visitors dominated the remainder of the first half, chances were few and far between as the Atleti back line effectively marshalled Messi and Suarez.
Oblak parried clear when Suarez finally worked a yard of space to shoot on target 10 minutes after the restart, before Messi's free-kick defeated the keeper and instead bounced away off the upright.
With the pressure building on the home goal, Messi charged into the area to fire straight at Oblak and then drilled wide when he looked destined to score.
Further smothering saves from Samuel Umtiti and Suarez followed, but Oblak was finally beaten by the latter in the closing stages.
Substitute Roberto delivered beautifully for Suarez to crash a header into the net, becoming the first domestic opposition player to score at the Wanda and earning Barca a deserved draw.
Messi had a last-ditch opportunity to take all three points with another free-kick, but his firm drive was gathered by Oblak as the full-time whistle blew.
