Leicester Tigers boss claimed it is "an embarrassment to the game" that received a yellow card in the Premiership final - which Bath edged at Twickenham.
The Midlands outfit were looking to end their era on a high. Head coach Cheika is leaving whilst club legends Cole and Ben Youngs were both playing their last ever games after trophy-laden periods with the Welford Road outfit.
Both appeared from the replacement bench, but Cole was yellow carded and sent to the sin bin for the final 10 minutes as his career ended on a low. He attempted to charge down Finn Russell but made contact with the Bath fly-half and was penalised by referee Karl Dickson.
Cheika bemoaned the performance of the referee, claiming they "got nothing back" despite their perceived dominance a scrum time. the Leicester boss was also stunned that Cole received his yellow and stated that rugby will be embarrassed by such decisions.
The Australian told : "I thought we got nothing back from the referee at all and we kept on coming back and turning up. We scored three tries to two, that tells you something. I’m super proud of the players.
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“[I was] just upset with the decisions on the field. The scrum we were dominating – obvious ones, not 50-50s. The scrum we were dominating completely. A couple of head knocks, a couple of balls where we could have got penalties there easily. [We had] two men in the bin, [and] the second one [was] for nothing, really. It is embarrassing for the game that a player goes off for that. I’m allowed to be angry, but I got over it and am happy to accept the result now and get on with the next thing.”
Pressed further on the yellow card incident he said: “For me personally, if the game is sending players to the sin bin for that, or even penalising [them]…What do I say, mate? If I say something, I’ll get myself in strife and take away from the champion team who have led the competition and deserve to win the competition."
Cole had only been on the pitch for seven minutes, running on to a heart-warming reception, and had looked to charge down the Bath 10. His yellow meant that the Tigers were up against it to mount a comeback, even if they kept scoring points with a man less.
Leicester came up with two late tries to set-up a grandstand finish, but ultimately came up short as Bath won their first Premiership title for nearly 30 years. Thomas Du Toit and Max Ojomoh came up with their tries as Youngs and Cole missed out on a champion farewell.
"It hurts, but I'm proud," Youngs told Radio Leicester. "My emotions are all over the place right now. You have so much ambition to win today and you don't, and you have that to deal with as you try to take it in. It's a funny old feeling."
Meanwhile Bath skipp Ben Spencer said: "The future of this club is unbelievably bright, no matter who pulls on the shirt it's next man in. The work ethic is second to none. As long as we keep our feet on the floor and keep on wanting to get better, the sky is the limit for this team."
He added: "The amount of hard work this has taken to turn the ship around... we were bottom three years ago. I can't credit [Johnan Van Graan] enough, he's been absolutely brilliant. If we [understand] there's always an opportunity to get better then I'm really excited for the next couple of years."