Atleti have never been beaten or even conceded a goal against their neighbours in Simeone's seven years in charge, and another positive result appeared inevitable once Griezmann broke the deadlock with his side's first shot of the match in the 27th minute.
Saul pounced to score a second before the break, which proved enough for Atleti, although a relatively comfortable outing was spoiled by the loss of Diego Godin to a thigh problem.
Only Atleti boast a better defensive record than Getafe in LaLiga this season, and the visitors kept Simeone's side comfortably at bay until shortly before the half hour, when Griezmann collected Thomas Partey's pass and rifled home his 10th league goal of the season from just inside the box.
David Soria saved Atleti's next shot on target, Nikola Kalinic shooting straight at the keeper after a fine move involving Thomas and Lucas Hernandez, but Saul reacted quickest to sweep home the rebound and double the lead.
Getafe applied some pressure after the break without managing to trouble Jan Oblak, and Atleti youngster Victor Mollejo, on as a substitute for Kalinic, blasted narrowly over the bar within moments of his introduction.
Oblak made a good save to deny Samu Saiz, while Rodrigo made a mess of a late one-on-one, before Djene Dakonam and Leandro Cabrera were sent off in the dying minutes for earning second yellow cards.
What does it mean? Pressure back on Barca in title race
Atletico's win, their 12th of the season in LaLiga, sees them close to within two points of leaders Barcelona, having played a game more.
Ernesto Valverde's side will know they cannot afford many slip-ups as long as Atleti's form persists, making Sunday's trip to neighbours
Girona a potentially key encounter.
Getafe remain sixth in the table, but will drop below
Real Betis if Quique Setien's side beat Athletic Bilbao.
Thomas keeps Atleti ticking
Thomas is not often lauded for his playmaking abilities, but his impact on both goals showed just how creative he can be at the heart of midfield.
A day to forget for Djene
Having lost Griezmann for the opening goal, Djene compounded a fairly miserable outing with a careless foul on Thomas Lemar in the 88th minute that earned him a second booking.
What's next?
Atleti face a difficult trip to Real Betis next Sunday, six days before the Madrid derby at the Wanda Metropolitano. Getafe meet
Valencia in the
Copa del Rey on Tuesday before resuming league duties at
Levante four days later.
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