Staunton scored 2-10 in Swinford as Mayo claimed a 4-17 to 1-13 victory, ending Armagh’s unbeaten record.
With Staunton on top form, Mayo led by 3-9 to 0-7 at half-time against the visitors.
The sides were level at 0-1 apiece in the early exchanges before Staunton struck for Mayo’s opening goal.
With 13 minutes played, Staunton netted again to hand Mayo a 2-4 to 0-1 lead and Sarah Rowe added a third goal to effectively put the tie to bed.
Armagh opened the second half with a goal from Caroline O’Hanlon but Mayo added another before the finish, with Aileen Gilroy on target.
Mayo preserved their 100% record with this victory and already, a place in the top four looks all but secured.
Elsewhere, Dublin maintained their good start to the season by collecting a third victory from four outings.
The Sky Blues gained some measure of revenge for last year’s All-Ireland final defeat to Cork by scoring a 3-10 to 1-13 victory over the Rebels at Parnell Park.
The opening score of the game was a Cork goal, scored by Orla Finn, and the Leesiders led by 1-7 to 1-3 at the break.
Lyndsey Davey scored a brilliant Dublin goal to keep the hosts in touch and they kickstarted their revival with a goal early in the second half from Amy Ring.
Carla Rowe scored a third Dublin goal to hand Greg McGonigle’s side a 3-7 to 1-10 lead, and they would hold out until the finish to collect maximum points.
Rowe finished with 1-4 for Dublin while Finn ended as Cork’s leading scorer with a haul of 1-6, including five pointed frees.
Monaghan collected their first victory of the campaign as Ulster rivals Tyrone were sent packing on a 1-15 to 0-4 scoreline in Drumhowan.
Therese McNally’s first-half goal helped Monaghan to lead by 1-9 to 0-3 at the break, on a day when Ellen McCarron and Cora Courtney scored five points each for the Farney girls. Tyrone managed just a single point from play in the entire game, scored by Maria Canavan, and remain firmly anchored to the bottom of the table, without a point from four outings.
Also in Division 1, Kerry defeated Galway by 2-15 to 2-8 in Finuge.
Roisin Leonard netted a 20th-minute penalty for Galway but Kerry had the lead at half-time as Bernie Breen netted seven minutes before the break.
Kerry were 1-7 to 1-5 ahead at the halfway point but Galway struck the first big blow of the second half when Mairead Seoighe netted.
Sarah Houlihan’s goal in the 46th minute had Kerry 2-12 to 2-6 clear, however, and they held on to move into the play-off picture.
In Division 2, there were victories for Donegal, Westmeath, Clare and Kildare, as Waterford, Wexford, Roscommon, and Tipperary claimed victories from their Division 3 outings over the weekend.
In Division 4, Derry, Longford, and Limerick claimed wins over Carlow, Wicklow and Antrim respectively.