Burton began the brighter with their best chance falling to Chris O'Grady, but his shot was blocked on the line.
For the second game running, the home side switched off in first-half stoppage time and an unmarked Connolly headed home to put Wigan ahead.
The 19-year-old defender, who joined on loan from Everton before the game, nodded in his second after the break.
England Under-19 international Connolly's move - which is for the remainder of the season - was confirmed by Wigan in time for him to make the trip to the Pirelli Stadium.
The Liverpool-born teenager, who spent a month on loan at Barnsley last season, came through Everton's academy.
And he helped his new club to their first victory over Burton Albion and back-to-back wins following victory in the FA Cup against Nottingham Forest.
The Brewers have now lost their past four home games in the Championship and sit just one point above the relegation zone.
Wigan remain 23rd, but are now just three points behind them.
Burton manager Nigel Clough:
"It's a frustrated dressing room because we have given away two of the softest goals we have given away all season.
"The timing as well is difficult. I think that is three in recent games just before half-time, in added time, and that is something that has to be addressed.
"Both penalty areas, which is where it really counts, they were better than us today."
Wigan manager Warren Joyce:
"It's a great win but it's just three points and there is still a long way to go in the season.
"Today, Burton could have scored a couple of goals but the way the defenders put their bodies on the line consistently you sometimes earn a little bit of luck when you are that brave."
On new signing Callum Connolly: "He's a player I've known about for a long time.
"Obviously when you are looking at the make-up of players and looking at their attributes and qualities, we are not a big side and its important in this league that you have got full-backs that can head the ball as well.
"Every credit to the kid today, enthusiasm and bravery gets us the points."
Match ends, Burton Albion 0, Wigan Athletic 2.
Second Half ends, Burton Albion 0, Wigan Athletic 2.
Offside, Wigan Athletic. Sam Morsy tries a through ball, but William Grigg is caught offside.
Damien McCrory (Burton Albion) wins a free kick in the defensive half.
Foul by Michael Jacobs (Wigan Athletic).
Corner,  Burton Albion. Conceded by Dan Burn.
Attempt missed. Kyle McFadzean (Burton Albion) header from the centre of the box misses to the left. Assisted by Matthew Palmer with a cross following a corner.
Corner,  Burton Albion. Conceded by Shaun MacDonald.
Attempt blocked. Tom Naylor (Burton Albion) right footed shot from outside the box is blocked.
Attempt blocked. Jamie Ward (Burton Albion) right footed shot from the left side of the box is blocked. Assisted by Chris O'Grady.
Attempt missed. Matthew Palmer (Burton Albion) right footed shot from the centre of the box is too high. Assisted by John Brayford with a headed pass.
Attempt saved. Chris O'Grady (Burton Albion) left footed shot from outside the box is saved in the bottom right corner.
Delay over. They are ready to continue.
Substitution, Wigan Athletic. Craig Morgan replaces Max Power because of an injury.
Substitution, Burton Albion. Tom Naylor replaces Luke Murphy.
Substitution, Burton Albion. Damien McCrory replaces Lloyd Dyer.
Delay in match Max Power (Wigan Athletic) because of an injury.
Attempt blocked. Kyle McFadzean (Burton Albion) right footed shot from the centre of the box is blocked.
Jordan Flores (Wigan Athletic) is shown the yellow card for a bad foul.
John Brayford (Burton Albion) wins a free kick on the right wing.
Foul by Jordan Flores (Wigan Athletic).
Substitution, Wigan Athletic. Jordan Flores replaces Yanic Wildschut because of an injury.
Delay over. They are ready to continue.
Delay in match Yanic Wildschut (Wigan Athletic) because of an injury.
Callum Connolly (Wigan Athletic) is shown the yellow card for a bad foul.
Delay over. They are ready to continue.
Delay in match Marvin Sordell (Burton Albion) because of an injury.
Delay in match Callum Connolly (Wigan Athletic) because of an injury.
Marvin Sordell (Burton Albion) wins a free kick in the defensive half.
Foul by Callum Connolly (Wigan Athletic).
Corner,  Burton Albion. Conceded by Jake Buxton.
Attempt blocked. Matthew Palmer (Burton Albion) left footed shot from the left side of the box is blocked. Assisted by Jamie Ward.
Foul by Marvin Sordell (Burton Albion).
Stephen Warnock (Wigan Athletic) wins a free kick in the defensive half.
Foul by Chris O'Grady (Burton Albion).
Callum Connolly (Wigan Athletic) wins a free kick in the defensive half.
Jamie Ward (Burton Albion) is shown the yellow card for a bad foul.
Foul by Jamie Ward (Burton Albion).
Max Power (Wigan Athletic) wins a free kick in the attacking half.
Corner,  Burton Albion. Conceded by Shaun MacDonald.
Callum Connolly scored twice on his debut as Wigan narrowed the gap to fellow relegation rivals Burton Albion.