Ronald Koeman's position as Barcelona coach has been repeatedly called into question after a poor run of form, with the result at Wanda Metropolitano making it one win in Barca's past six games.
Barca could have lifted some of the weight off Koeman's shoulders on Saturday, but Suarez set up Thomas Lemar for the opener and then fired in a second to put Atleti in control at half-time.
That is how the scoreline stayed as Diego Simeone's men inflicted a first league defeat of the season on their opponents to move level on points with Real Madrid at the top of the table.
Koeman was watching from the stands as he served the second of a two-game touchline ban and saw his side fall behind to a well-worked Atleti goal with 23 minutes played.
Suarez was picked out by Joao Felix and then played a first-time pass into the path of Lemar for the France international to thump past Marc-Andre ter Stegen.
Former Barca man Suarez curled wide with only Ter Stegen to beat soon after, but he made amends by picking his spot when played in by Lemar at the end of a swift counter one minute before the break.
Philippe Coutinho, having earlier volleyed narrowly wide, wasted a glorious chance to pull one back when failing to get the better of Jan Oblak with an hour on the clock.
Joao Felix and Barca substitute Ansu Fati exchanged shots on target at either end in the latter stages but no further goals arrived in the Spanish capital, potentially bringing down the curtain on Koeman's short-lived tenure.
Despite being given the public backing of Joan Laporta earlier in the day, widespread reports from Spain suggest this could well have been Koeman's final match in charge of Barca.
The ex-Netherlands boss may have had to work with one hand tied behind his back this term, but not even he can justify his side's shambolic defending – and toothless attacking display – once again here.
While Barca has a fortnight to stew on this defeat, Atleti's title hopes have been revived thanks to its first successive home league wins in this fixture since a run of three victories between March 2008 and February 2010.
It remains to be seen whether Koeman will still be in charge when Barcelona hosts Valencia on 18 October (AEDT). Atleti travels to Granada on its return to action in two weeks' time.
Bites from Atlético Madrid 2-0 Barcelona:
Atletico Madrid have not lost any of their last four meetings with Barcelona in LaLiga (W2 D2), their best unbeaten run against them in the competition since five games between September 2004 and December 2006 (W3 D2).
Atlético de Madrid have gone to half-time with +2 goals difference against Barcelona for the first time in LaLiga in the 21st century.
Atlético de Madrid have just lost once in their 40 home LaLiga games against Barcelona scoring, at least, two goals (33V 6E). It was in April 1997 (2-5).
Atlético de Madrid have kept a clean sheet in three consecutive games against Barcelona for the first time ever in LaLiga history.
Atletico Madrid's Luis Suarez has scored against 31 of the 31 opponents he has faced in LaLiga (172 goals in total).
Atlético de Madrid goalkeeper Jan Oblak has kept a clean sheet in his last three matches against Barcelona in LaLiga, saving all nine shots on target he faced. The Slovenian has become the first goalkeeper to keep a clean sheet in three consecutive games against them in the competition in the 21st century.
Since his debut on 25th October 2014 against Real Madrid, when he also provided an assist, only Lionel Messi has more assists (91 in 235 games) than Atlético’s Luis Suárez (73 in 231 games).
Thomas Lemar has scored two goals in his last two LaLiga games for Atlético de Madrid, as many as in his previous 78 games in the competition.
Atlético’s Thomas Lemar has scored and assisted in a single LaLiga game for the first time ever in the competition (85 games played).
Barcelona coach Ronald Koeman has not won any of his three managerial games against Diego Simeone in LaLiga (D1 L2), more than against any other manager without a win in the competition (also three against Álvaro Cervera -D2 L1-).