The 2-1 win not only put the Toffees' top-four challenge back on track but was also a morale booster after their 4-0 midweek derby defeat to Liverpool.
All was far from rosy at half-time on Saturday as an unusually hesitant Everton trailed 1-0 to Leandro Bacuna's 34th-minute opener.
But the arrivals of Steven Pienaar and Steven Naismith from the bench changed the game, with the latter equalising in the 74th minute before Mirallas had the final word.
Leighton Baines is regarded as Everton's free-kick specialist but the full-back stood and watched as Mirallas placed the ball just inside the post from 30 yards with five minutes to go.
"I practise free-kicks every day to try to get them right on the day of the game," Mirallas said on evertonfc.com.
"I'm very happy because my father and my mother came to see the game and it was a good goal."
Injuries have taken their toll on Everton and, in the absence of Romelu Lukaku and new signing Lacina Traore, Mirallas was asked to play as a makeshift striker.
The winger has experience of the role from his Olympiacos days but had endured a frustrating afternoon prior to the spectacular denouement.
He said: "I played as a striker. It's not my best position but I worked for the team. It was a nice goal and it was very important to win because in the last game against Liverpool we played badly. It was a bad result and we needed a win."
Defender Sylvain Distin, who returned to the side earlier than expected from a hamstring problem, admitted the Liverpool result played a factor in their first-half performance.
He said: "I think maybe we were a bit cautious in the first half, and maybe we still had the Liverpool game in our minds. In the second half we started to play our game.
"We had most of the ball and they didn't look dangerous so it was frustrating to concede a goal but that's part of football. We spoke in the dressing room at half-time and we knew it wasn't good enough.
"We know it's going to be extremely difficult to finish in the top four but there's no reason not to believe and be positive."
Bacuna has shown a real eye for goal since joining Villa from FC Groningen in the summer and Saturday's confident finish gave him a fourth in the Premier League this season and second in as many games.
The full-back felt the result was rough on the away side, saying: "I think we defended well and had a great game, we were a bit unlucky with the first goal and the second goal was just a good free-kick."
Villa had turned their form around with four points from their previous two games to climb to 10th place, but the close nature of the bottom of the table means they are closer to Cardiff in 19th than they are to Southampton in ninth.
"We're 10th and want to go up, we're looking up," said Bacuna.
Villa's cause was not helped by captain Ron Vlaar being forced off 10 minutes from time due to injury.
Manager Paul Lambert said: "It's his hamstring so we'll just have to see how he is. Hopefully it's not too bad."
The Scot was largely positive about his team's display and felt they deserved to leave Merseyside with something for the second time in a fortnight.
Two weeks ago Villa drew 2-2 with Liverpool, and Lambert said: "It's not a bad sign if you can come to these two places and walk away disappointed.
"We're a young team and they've got to feel it, they've got to realise they're good enough to compete."