The Spitfires fell to defeat at the Silverlake stadium after goals from Newby and Crawford were enough to give the visitors all three points.
An early mistake from McDonnell gifted Newby an early goal before Shade was given his marching orders for a headbutt.
Francillette equalised moments after the dismissal but a late strike from Crawford took the game away from the Spitfires.
With just six minutes on the clock, the visitors seized the lead with their first opportunity of the game.
A miscued touch by McDonnell in goal presented the ball to Alex Newby, who calmly sidestepped him before slotting the ball into the empty net.
Another 13 minutes passed before the next chance arose, once again for the visitors, as Regan Linney’s curling effort drifted just wide of McDonnell’s goal.
Halfway through the first half, things went from bad to worse for the Spitfires as Tyrese Shade was shown a red card for a headbutt on Olsen.
However, the Spitfires responded almost immediately when Francillette levelled the score with a brilliant low volley, capitalising on a loose ball inside the penalty area.
Four minutes before the break, Humphries was cautioned by the referee for a trip on Lucas Weaver.
The score was level at the interval, with Francillette’s volley cancelling out Newby’s early opener.
The visitors had the first opportunity of the second half when Linney was played through on goal and seemed certain to score, but McDonnell stood his ground and produced a brilliant save with his right leg.
Banks was next to try his luck for Altrincham on the hour mark, but his looping header was comfortably saved by McDonnell.
With 24 minutes to play, Banks tried his luck again, this time from outside the area but his effort sailed high into the stand.
Moments later, the ball dropped to Weaver inside the area, and he seemed destined to score, but a brilliant block from Boutin denied his goal-bound strike.
In the 71th minute Altrincham regained their lead.
Crawford fired a low shot from outside the box which wrong footed McDonnell and nestled into the bottom corner.
Two minuted from time, Semenyo tried his luck with a low effort from 25 yards but Poss saved comfortably low to his left.
Altrincham then had two chances simultaneously to put the game to bed, the first was cleared off the line by Francillette before McDonnell palmed away a vigorous strike from outside the box.
In stoppage time, the visitors thought they had a third as Bickerstaff fired low past McDonnell but he was ruled offside by the linesman.
McDonnell was once again called into action as he denied Linney who was through on goal late on.
That proved to be the last action of the game as Altrincham head back up north with all three points.
Eastleigh: McDonnell (GK), Brindley, Humphries (Ryan 60′), Francillette, Maguire, Taylor (C), Shade, Hodson, Waruih, Boutin (Semenyo 78′), Maher
Unused subs: Scott (GK), Fernandez, Dotse, Underhill
Goals: Francillette (27′)
Yellow cards: Humphries (41′), Semenyo (80′)
Red Cards: Shade (24′)
Altrincham: Poss (GK), Banks (Cooper 88′), Baines (C), Osborne, Linney, Newby, Weaver (Reddin 78′), Crawford, Golden, Nuttall (Bickerstaff 64′), Olson (Jones 45′)
Unused subs: German, Randle, Pasiek
Goals: Newby (6′), Crawford (71′)
Yellow cards: Olson (25′)
Referee: Robert Massey-Ellis
Attendance: 1931 (82 away)
Report by Josh Jablonski