Ryan Strome had a goal and an assist, and Sam Carrick and Cam Fowler also scored for the Ducks. Mason McTavish had two assists.
The 22-year-old Dostal set a new career high for saves - topping his previous best of 35 set in his season debut on Dec. 12 -while making his sixth career start.
''I am speechless, it is unreal,'' Dostal said of beating the Oilers. ''Those guys are just something else. It was a privilege to play against them and it is even more exciting to get the W. It is a pretty nice moment.''
The Ducks won in regulation for the second straight game and just the third time this season.
''There have been times this year where maybe we have played well and didn't get rewarded,'' Anaheim coach Dallas Eakins said. ''There have been times like tonight with the amount of firepower these guys have that maybe we should have been on the other side. It is good for the group.''
Darnell Nurse, Ryan Nugent-Hopkins and Connor McDavid scored for the Oilers, who have lost three of four. Leon Draisaitl had two assists and Stuart Skinner finished with 13 saves.
''Too many individual mistakes from everyone - defensemen, forwards - and then we're maybe a little out of position after that where we're chasing it. Gotta clean that up,'' Draisaitl said. ''It's frustrating right now. But we can't hang our heads. Have to park it, learn from it, and we've got a big road trip coming up.''
Trailing 3-1 late in the second period, the Oilers pulled within one on the power play with 28 seconds remaining as McDavid put a puck in front and it pinballed around before Nugent-Hopkins shoveled in his 15th. McDavid extended his point streak to 12 games on the play.
Edmonton tied it on a two-man-advantage 2:06 into the third as Leon Draisaitl sent a pass through the seam to give McDavid a wide-open net to shoot for his league-leading 28th.
Anaheim regained the advantage 4-3 at 6:08 as Oilers defenseman Evan Bouchard not only gave the puck away but then screened Skinner as Klingberg floated in a long shot for his fourth.
The Oilers outshot the Ducks 23-2 in the third period, but Dostal shut the door.
''He was calm and having fun,'' Eakins said. ''He was very composed. He is part of our future.''
Edmonton started the scoring 5:07 into the game as Nurse fired a long shot from the point that eluded Dostal, Anaheim's third-string goalie.
Anaheim tied it with 5:23 to play in the first as Carrick had a wide-open net to tap in his first goal of the season thanks to a great setup by McTavish.
The Ducks took a 2-1 lead on the power play with 2:41 left in the opening period as McTavish dropped it back to Fowler and he fired a long shot past Skinner.
Anaheim took a two-goal lead 8:31 into the second as a terrible giveaway by Bouchard at the Ducks' blueline gave Strome a clear breakaway and he was able to snap in a shot for his eighth.
Edmonton came in with the league's best power play at 31.9% and went 2 for 6 against Anaheim, which had the league's worst penalty kill (67.3%). ... The Ducks were without G John Gibson (undisclosed), LW Max Jones (upper body), D Jamie Drysdale (shoulder), C Isac Lundestrom (fractured finger), C Derek Grant (lower body) and G Anthony Stolarz (lower body). ... The Oilers were missing LW Evander Kane (wrist) and C Ryan McLeod (ankle). LW Warren Foegele returned after missing 10 games with an undisclosed injury.
Ducks: At Los Angeles on Tuesday in the finale of a five-game trip.
Oilers: At Nashville on Monday in the opener of a two-game trip.
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