Dmitrovic drove an early spot-kick past fellow goalkeeper Jan Oblak, only for Suarez to inspire the comeback, his second also a penalty which he earned himself and then converted in the 89th minute.
Another late victory for Atletico means they pull seven points clear of second-placed Real Madrid and 10 ahead of third-placed Barcelona, with another game in hand over both still to come.
Suarez, meanwhile, moves level with Lionel Messi at the top of La Liga's scoring charts on 11 goals, the Uruguayan increasingly looking like a player that can lead his side to the title.
He had Marcos Llorente's pressing to thank for his equaliser but the second was almost entirely down to him, the 33-year-old chasing down a long punt downfield before tumbling over in the box.
Suarez chipped the penalty coolly down the middle.
Atletico were the victims of an upset in the Copa del Rey a fortnight ago when they lost to third-tier side Cornella, whose spirited run came to an end at the hands of Barcelona on Thursday.
Atletico's defeat by Cornella could have jolted their momentum. Instead they have responded with two wins in the league, against Sevilla and now Eibar, who stay 15th.