The 22-year-old winger, who has had to be patient since joining from Burnley in January, set up an early opener for Joe Westley and then added a goal of his own in the second half.
While the Dark Blues savoured an impressive win, Aberdeen's huge travelling support regularly vented frustration throughout what was a dismal afternoon for the visitors.
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Dundee boss Steven Pressley selected the same XI that started Monday's 1-0 defeat away to Celtic.
There were two enforced alterations to the Dons line-up that kicked off last weekend's 2-1 win over Hearts as Alex Briedel and Stuart Armstrong replaced injured midfielders Connor Ronan and Brad Lyons.
Bevan signalled his intent early on when he cut in from the left and drilled just wide from edge of the box.
The hosts made the breakthrough in the ninth minute when Westley met Bevan's inswinging cross from the right at the near post and steered home a brilliant cushioned volley from just outside the six-yard box.
The Dons were struggling to impose themselves and they had a major let-off when Westley was denied by Marius Muller after going clean through on goal and then cut the loose ball back into the path of strike partner Simon Murray, who fluffed his shot tamely at the keeper from a central position.
Dundee spurned another golden opportunity when Charlie Reilly blasted over after being teed up by Murray 15 yards out, with booing audible from sections of the Dons support exasperated at their team's meek start.
The visitors almost equalised with their first chance in the 34th minute when Dan Happe's header from Armstrong's corner was tipped on to the crossbar by Owen Goodman.
But two minutes later, Bevan went clean through and rolled his low angled shot agonisingly wide of the far post.
Aberdeen's frustration was underlined in the 38th minute when Stephen Robinson was shown a yellow card for protesting after Lewis Mayo was booked for a rash challenge on Simon Murray.
Bevan then saw another effort pushed out by Muller as Dundee went in with only a one-goal lead to show for their first-half superiority.
Robinson made a double change for the start of the second half as Ayoub Mouloua and Lewis Smith were sent on for Afeez Aremu and former Dee winger Tony Yogane.
But the bedraggled Dons fell further behind in the 56th minute when Charlie Reilly's cross from the left fell to Bevan at the far post and the unmarked winger took a touch before firing a firm shot beyond Muller from eight yards out.
Substitute Ashley Hay missed three good chances to extend Dundee's lead in the closing half hour, while Chris Cadden went closest to a counter for the Dons when he struck the bar in the 63rd minute.
