Hearts bounced back from the defeat at Hibernian which saw them relinquish their hold on the William Hill Scottish Cup at the first hurdle with Saturday's victory over the Dons, who have not scored against the men in maroon in eight matches.
Stevenson converted a 31st-minute penalty before departing at half-time following a clash of heads, his replacement Callum Paterson adding a second after 54 minutes.
Hearts - who moved to seventh in the Clydesdale Bank Premier League, 10 points behind leaders Celtic - play at St Johnstone next week and meet Dundee United, Kilmarnock and Ross County before the next Edinburgh derby on January 3.
"You just hope that we can go on a decent run now," said Stevenson, sporting a black eye following his interval substitution.
"We've got a run of games coming up that we're capable of winning.
"The league's so tight that you go on a decent run and the thing totally flips on its head.
"We look forward to next week now. It will be a hard game at St Johnstone, but it's one we're capable of winning."