The La Liga leaders will be without Costa for Saturday's clash at home to Villarreal after the Spain international suffered a hamstring injury in their UEFA Champions League draw with Barcelona.
Simeone said injuries were to be expected late in the season, but urged his side to find the answers without a man who has scored 25 league goals this campaign.
"It's normal. This is football and players are not machines," he said.
"We're not going to make any excuses because we're delighted to be doing what we do. We are a team and we have to find solutions as a team.
"Barcelona and Madrid might have more players in reserve but neither of them have two Cristianos (Cristiano Ronaldos) or two Messis (Lionel Messis)."
Simeone is wary of a Villarreal side sitting seventh and chasing a UEFA Europa League place this season.
The Argentine said fatigue would play no part despite his team having played their quarter-final first leg against Barcelona on Tuesday.
"The lads have all recovered and we've got tonight to rest," Simeone said.
"Tomorrow's game is an important one for us, against a difficult rival who play a positional game similar to ourselves and have a variety of combinations to their game from midfield to attack."
With Costa missing, Diego may be given a rare start at the Vicente Calderon.
The 29-year-old Brazilian attacker came on for Costa against Barca and Simeone was impressed by the former Werder Bremen and Juventus man.
"I'm really happy with how Diego is progressing. It's normal that after spending two years away from us, he's found it hard to reclaim his place," Simeone said.
"You cannot be a competitive side with eight top class players – you need at least 11 and if we find ourselves in that position now it's because of the players who are waiting in the background and who have had less opportunities."
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