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."
Stevenson assumed spot-kick duties after captain Marius Zaliukas followed up his successful penalty at Inverness with a miss against Celtic.
The 28-year-old roared with relief after confidently converting from 12 yards for his first Hearts goal since returning to Tynecastle in August.
In January, Stevenson departed, frustrated at wage delays but he was aware there would be further trouble ahead when returning from Ipswich.
"It had been a long time since I had scored, coming back," Stevenson said.
"A wee bit more added pressure, but it was just nice to see it hit the back of the net."
On his withdrawal, he added: "I got a knock on the head at Dundee United and tried to come back out. It didn't help the situation.
"It was probably the right thing to do to come off. I was just disappointed to come off after a good half."
Aberdeen manager Craig Brown again bemoaned the absence of eight players through injury, but is pleased his squad return to action quickly, with the Scottish Cup fourth-round replay at Motherwell. The winners play Hibs in February.
Brown said: "I'm happy we've got a chance on Tuesday. It will be a difficult cup tie against Motherwell.
"Some of the substitutions were made with Tuesday in mind because it was getting towards the end of the game. The guys we replaced were older players, like Russell Anderson.
"That's not to say we conceded defeat. We said we'd use our resources to the best and fresh legs might turn it.
"Each one of the substitutes had a great chance.
"But credit to Hearts. Hearts are in a false position in the SPL. They're a good side and getting together, I think."
Brown was also asked about Ryan Fraser, the 18-year-old who has turned down a new contract at Pittodrie.
The Dons boss said: "He's a smashing wee guy and he just said he wanted to leave it at the moment. He's adamant it's not a question of money."