Connor McDavid had a team-high five shots on goal, Cam Talbot made 31 saves on 32 shots and Leon Draisaitl won 76 percent of his faceoffs.
Edmonton went 0-for-3 on the power play and 2-for-2 on the penalty kill.
"I'm not disappointed in the effort," said Oilers Head Coach Todd McLellan. "I'm disappointed in the outcome.
"Both teams checked and defended well. There wasn't much that was given up."
Mike Smith made 28 saves in his shutout performance, robbing the Oilers on several chances.
"Both goaltenders made some tremendous saves," continued McLellan. "We had some real good looks and Smith made some saves but so did they early in the game on the power play and Cam made saves."
It was a back-and-forth first period in the fourth instalment of 2017-18's Battle of Alberta, as neither side could find twine but plenty of chances were had.
McDavid - coming into the game with 31 goals and 84 points on the season, four back of NHL scoring leader Nikita Kucherov - had his opportunities.
The Oilers Captain was flying early and driving to the net on several occasions. In one instance, McDavid ripped up ice and put the puck on netminder Smith. Pontus Aberg crashed the net looking for the rebound but Calgary was able to avert any danger.
McDavid later busted in on a partial breakaway but lost the handle of the puck after a stick-check from behind.
At 14:05 of the period, the Oilers thought they had scored after Zack Kassian whacked away in the crease. The goal was immediately called off, as the whistle was blown before the puck crossed the goal-line.
"It wasn't reviewable because the whistle was deemed to have gone before it crossed the line," said McLellan of the play.
The Flames opened the scoring in the second period, getting a marker at 5:18 from Gaudreau.
The forward found himself all alone in front of Talbot. Gaudreau shifted to his backhand and flipped the puck over Talbot's shoulder, scoring his 22nd goal and 80th point of the season.
"I thought we made one mistake on a faceoff coverage," said McLellan. "We cheated a little in the D-zone and it was in our net. Other than that, the effort on the road was pretty good."
Edmonton got two power plays in the period, nearly converting on one of them when Ryan Strome almost tallied off a shot in the slot. Somehow, the Flames were able to clear the puck once again and kill both penalties.
With less than 10 seconds remaining in the middle stanza, Smith intercepted a cross-crease feed from Jesse Puljujarvi intended for Anton Slepyshev to keep Edmonton off the board.
Smith continued his strong play in the final 20 minutes when the Oilers looked for the equalizer. The keeper stoned Ryan Nugent-Hopkins on a 2-on-1 with McDavid, then McDavid was thwarted on yet another individual rush.
"You have a guy like that who's athletic and big, it's tough to beat them when they're on," said Nugent-Hopkins of Smith. "Obviously he was on tonight."
The Oilers got a chance on the power play with under two minutes to play but couldn't convert.
Edmonton returns home for a match against the San Jose Sharks Wednesday.