The Serb followed up a third-round victory over holder Grigor Dimitrov earlier in the day with a 7-5 4-6 6-3 victory over Canadian Milos Raonic.
But after the pause, the pair came back out, with Federer finally securing the first break of the match on his seventh chance to take a 4-2 lead over his longtime rival.
Top-ranked Simona Halep plays later, trying to reach the finals for the third time.
By entering the final, Čilić would jump to fifth place in the individual ATP ranking, and by winning his second title in Cincinnati, he would skip Alexander Zverev to be the fourth best tennis player in the world.
Federer has won an unprecedented seven Cincinnati titles, beating Djokovic three times in championship matches.
A wide forehand off the next point saw the Belgian's situation worsen, and Federer took advantage, sending down his eighth ace of the set to take control, before another netted forehand rally ball saw Goffin succumb.
In June 2016 Djokovic beat Andy Murray to win a first French Open title and with it become the first man since Rod Laver in 1969 to hold all four Grand Slam titles at the same time.
Only a couple of hours later, Djokovic stopped Milos Raonic 7-5, 4-6, 6-3, improving his career record to a ideal 9-0 against the Canadian.
Rain wreaked havoc on Thursday's schedule, leading to several players having to take the court twice on Friday.
Federer won their only previous final here in 2015 in straight sets.
"I just need to get out there and really fight, I was telling myself to fight", the two-time Wimbledon champion said.
She'll face Kiki Bertens, who advanced to her first Western & Southern final by wearing down No. 8 Petra Kvitova 3-6, 6-4, 6-2.
"I feel like she missed more than me today", Halep said of Barty.
Because of the tournament's many weather delays, Halep will be back on the court later on Friday to play Ukrainian Lesia Tsurenko in the quarter-finals. "So before the match, I'm less nervous than I used to be".
The Serb, a 13-time Grand Slam victor, claimed a memorable Wimbledon title in July - a victory that marked the culmination of a two-year rise and fall from the very top of the game.
She's expecting a lot of attention at the U.S. Open, where she reached the final past year and lost to Sloane Stephens - her best showing in a Grand Slam event.
After his passing shot finished the 2-hour, 30-minute match, Djokovic led the crowd in cheers and pumped his fist, ending a long day with an impromptu celebration.
"My legs were pretty heavy", Kvitova said.
"And that's how I was - in some points I didn't really risk that much and just put it to the court".
"I got my energy back for the third set, a momentum shift was good for me". Tonight I was really close.