Haaland made his first appearance since fracturing his foot in the defeat at Aston Villa on 6 December when he was introduced as a 71st-minute substitute at Etihad Stadium.
However, while the Norwegian's major involvement was to draw laughs from team-mates Bernardo Silva and Mateo Kovacic on the bench when he mis-kicked what was intended to be a shot on the turn, Alvarez turned 24 in style.
The Argentine took the man-of-the-match honours by becoming just the 10th player to score two goals or more on their birthday in a Premier League game.
Both efforts came in the opening period. Matheus Nunes chipped a superb cross into the six-yard box for Alvarez to head home in the 16th minute.
And just six minutes later, Alvarez profited from a brilliantly worked training ground free-kick routine as Kevin de Bruyne shaped to curl a cross into the box but instead rolled a perfect 20-yard pass which just needed the first-time finish.
Burnley boss, and former City captain, Vincent Kompany would have been pleased with the way his side kept battling until the end.
They were at least able to grab an injury-time consolation as defender Ameen Al-Dakhil bundled in their first goal in seven visits to this stadium since 2018.
They were ultimately powerless, though, to stop second-placed City's eighth successive win in all competitions - and 13th consecutive victory against second-bottom Burnley.
And boss Pep Guardiola believes his side need a significant winning run if they are to retain the title.
"With this Liverpool that we have faced for eight years, and Arsenal, with Mikel [Arteta] who has settled his squad, I don't have the feeling that they're going to drop much points," he said.
"We have to win as many games as possible and, at the end, we will see what happens."
Praise for City backroom hero Vicens
Once he had expressed delight at Alvarez's second goal, Guardiola turned to his bench and located Carlos Vicens.
A decent youth player from the Balearic Islands, Vicens spent his career in the Spanish lower leagues before eventually turning to coaching.
Remarkably, he initially came to City on work experience in 2017 and was eventually appointed assistant coach to the under-12s and under-13s teams before working his way up to under-18 level.
Vicens is now City's set-piece coach and Guardiola appreciated the work that would have gone into an initial idea being executed to perfection in a Premier League game.
Of course, the plan is only one aspect. You also need players with the ability to execute and in De Bruyne City have one of the very best in the world.
Starting for the first time since suffering a recurrence of the hamstring injury he initially sustained in the Champions League final at Burnley on the opening day of the season, De Bruyne took deception to 'Traitors' levels. He stared into the penalty area and raised his arms to indicate the kind of cross that was coming before quickly changing his body shape to roll his pass straight into Alvarez's path.
With City in control after that strike, it was just a case of waiting for Haaland's return to action.
The forward was cheered when he came out for his warm-up and again when he replaced De Bruyne.
There was a familiar menace about his running style and the way Haaland attacked the ball. But after being ignored by Jeremy Doku as he ran into the box, then came the moment that will doubtless be the topic of much dressing banter.
Once the inevitable rust disappears, it should not be too long before Haaland makes the right contact and starts scoring again.
"It has been really good without him for the last two months," added Guardiola. "But with him we are a better team and we are really pleased that he is back."
Burnley relegation concern continues
Without Everton's 10-point deduction, they would already be eight points adrift and Forest would be just above the relegation zone.
Kompany knows more than most what to expect in this stadium and will be disappointed his players were not alive enough to react to either Nunes' impudent cross or De Bruyne's clever free-kick.
Given Burnley had conceded 28 times since they last scored against City, victory was implausible for the Clarets - who have won just three times this season -once they went behind.
But there was never any real belief they could make a proper game of it, with even Al-Dakhil's first goal since joining Burnley 12 months ago from Belgian outfit Sint-Truiden coming as something of a surprise at the end.
"We played one of the best teams in the world," Kompany told BBC Match of the Day.
"In these games it's about being able to build chances and momentum. I felt if we had the last pass coming off, if we put it in the back of the net, it would help of course.
"We had our moments. I can dissect it as much as I want, but we have to move on to Fulham now."