The two sides meet again on Sunday night at Coventry's Skydome Arena with a place in next weekend's finals weekend in Nottingham at stake.
Cloutier had warned his players to stay out of the penalty box before the game at Hull Arena but it was two powerplay goals which did for his side, who were unable to convert any of their own - including two 5-on-3 situations.
Both goaltenders were on good form and the deadlock was not broken until 33.42 when the visitors' powerplay unit came up with the goods and Luke Fulgum reached the 50-goal milestone for the season.
Less than a minute later, however, and Stingrays were back on level terms after top scorer Konstantin Kalmikov fired home at 34.12 for his 37th of the season.
The intensity from both sides made for a pulsating first two periods and, while that faded slightly in the third, there was still little to choose between top and bottom of the regular season table.
Coventry ensured they took a one-goal advantage into the second leg, however, when Greg Chambers struck on the powerplay at 47.19.
"We are still very much in it," said Cloutier afterwards.
"Ideally we wanted to either be tied or even one-up after the first leg but we find ourselves one down.
"But we're now a confident group and we've proved we can beat Coventry. We've got top go there and take it a shift at a time and see where that gets us.
"I thought it both sides played good hockey and we showed a lot of intensity - the players didn't need much firing up for this one. We had opportunities to go ahead when we had a 5-on-3 but it didn't happen but we'll go there and give it everything because we've got nothing to lose."
Sheffield Steelers fired a late goal in Cardiff to give themselves hope of progressing to the play-off finals weekend in Nottingham.
Jeff Legue's 59th-minute strike saw Steelers return from South Wales on the back of a 4-2 defeat in the first leg of their quarter-final clash with Cardiff Devils.
Steelers will have an opportunity to overturn the two-goal deficit when the two sides meet in the second leg at Sheffield Arena on Sunday night (6pm).
The home team who opened the scoring with a strike from Phil Hill at 12.35, but the visitors levelled through Mike Sgroi's fifth of the season at 15.18.
Two third period goals - the first on the powerplay from Mark Smith at 25.24 and another from Max Birbraer at 38.30 - then put Cardiff into a commanding position which they strengthened through Matt Miller's 31st of the season at 56.55.
But Legue's 40th goal of the season at 58.53 gave Steelers' travelling army of fans something to cling to ahead of Sunday's rematch.
Cardiff Devils 4 Sheffield Steelers 2
Belfast Giants 4 Newcastle Vipers 1
Edinburgh Capitals 4 Nottingham Panthers 4
Hull Stingrays 1 Coventry Blaze 2