Goals from Lawrence Shankland and Landry Kabore in the space of seven first-half minutes put the hosts 2-0 up by the interval.
Kabore added a second before substitute Tomas Magnusson bagged his first goal for the club.
Hearts, held by St Mirren in midweek, took advantage of second-placed Celtic's involvement in Sunday's League Cup semi-final against Rangers.
Derek McInnes's team remain undefeated in 11 league games this season, while back-to-back defeats leave Dundee second-bottom of the pile.
The home side opened the scoring in the 31st minute when Harry Milne sent a great ball into the box for Shankland to drill into the bottom right corner for his 10th goal of the season.
They doubled their lead in the 38th minute when Milne's corner from the left bounced around the box and fell to Kabore, who slammed into the net from close range to open his account for the club.
Kabore put Hearts 3-0 up in the 56th minute and substitute Magnusson made it four late on.
Celtic, under new manager Martin O'Neill following the shock departure of Brendan Rodgers, remain the bookies' favourites to win the Scottish Premiership.
But Hearts are in pole position to end the Celtic-Rangers domination of Scottish football.
No team outside the Glasgow giants have won the Scottish Premiership since Aberdeen in 1985.