Albion came back from 13-0 down early in the first half and with a minute of normal time remaining the score at Butts Park Arena was level at 19-19.
But Coventry, who finished third last season, won the game with a late converted try by replacement Jacques Le Roux.
Initially, it looked like referee Phil Davies had disallowed the try for a forward pass but, after consulting with his touch-judge, he awarded the score to break Albion’s hearts.
It was Plymouth’s third one-score defeat this season, having lost 16-15 to Rosslyn Park and 24-21 at Hull Ionians, where the home side were awarded two penalty tries.
“The scoreline says what it is and we have to accept it,” said a disappointed Dawe, whose team did out-score Coventry on tries.
“The guys fronted up well. They put the effort in, but for a couple of cheap penalties we could have won the game.”
On Coventry’s controversial winning score Dawe added: “It had to be the referee’s call as he was the one closest in line with the attack. He didn’t think it was a try, but he asked for clarification and it was given. Like I said, we just have to accept it.
“What we think is irrelevant. He penalised us in their 22 two or three minutes earlier when we had pressure and field position and that’s what probably cost us the game.”
He added: “This one was always going to be a big game coming so early in the season, away from home against a good team.
“Yet, but for a couple of decisions that were in our control and a couple of sloppy penalties it could have been a much closer game.
Saturday was the third time this season Albion have gone two scores down early on.
“We don’t give teams starts, they have scored them,” said Dawe. “You have to give credit where credit is due.
“We messed up two kick-offs in the first half on Saturday and Coventry profited.”
The big positive for Albion at the weekend was their set-piece, which was outstanding.
“We pride ourselves on that side of our game,” said Dawe. “We don’t want to take too many backward steps.”
Forwards Herbie Stupple, Nile Dacres, Dan Collier and Ed Holmes also stood out with their strong ball carrying.
“There’s always good performances from a team that fronts up,” said Dawe. “We stand and fall together as a team. There was a collective disappointment in not winning the game.”
The one worry for Albion was an injury to Australian winger Ethan Ford, who had to go off early in the second half after scoring his side’s second try.
“He damaged his knee just before he scored that try,” said Dawe. “He’s a tough cookie and hopefully he’ll be back sooner rather than later.”
With fellow wingers Robin Wedlake and Harrison Cully also out injured and Seti Raumakita away with the Royal Navy, Albion will be hoping Ford’s injury is not that serious, with a tough home match against Esher coming up this coming Saturday.