In Group 1, Armagh had to settle for a draw against Tipperary at FBD Semple Stadium, while 2022 and 2023 runners-up Kerry got the better of Waterford in Group 2 to advance.
In Group 3, Cork followed up on their impressive Round 1 victory over Laois with a home success against Galway at MTU.
In Group 4, meanwhile, Mayo bounced back from their Round 1 loss to Dublin with a home victory over Kildare.
Two goals in three minutes from Anna Rose Kennedy and Aisling Moloney helped Tipperary bounce back and claim a 2-4 to 0-10 draw against Armagh.
Hannah O’Donoghue and Danielle O’Leary struck goals in the opening half for Kerry as they built up a 2-10 to 0-3 interval lead against Waterford, with Emma Dineen and Siofra O’Shea finding the net for the All-Ireland runners-up after the break as they cantered to a 4-13 to 0-9 win.
Mayo were also big winners to spoil Diane O’Hora’s homecoming as Kildare manager, with Liam McHales’s charges running out 2-15 to 0-5 winners thanks to goals from the Cafferky sisters Lisa and Sinéad.
A goal from Hannah Looney was key for Cork as they defeated Galway by 1-12 to 1-10 to book a place in the last eight. Galway led by 1-7 to 0-7 at the interval thanks to a goal from Olivia Divilly.
Meanwhile, the last eight for the quarter-finals of the TG4 All-Ireland Intermediate Championships were confirmed on Sunday.
There were victories for Westmeath, Wexford and Leitrim, while Monaghan and Offaly drew 0-10 apiece.
Goals from Vicky Carr and Rachel Dillon helped Westmeath defeat Cavan by 2-12 to 2-6, with Ally Cahill getting the goals for Cavan.
Wexford defeated Antrim by 3-22 to 0-4 thanks to goals from Tara Doyle, Grace Donovan and Caitriona Murray.
Michelle Guckian (0-6) and Ailbhe Clancy (0-4) were on target for Leitrim as they defeated Wicklow by 0-17 to 0-10.
A late point from sub Sarah Capaldi from a free levelled the game and ensured Monaghan go through to the quarter finals after drawing 0-10 apiece with Offaly.
Down, Clare, Wexford and Leitrim finish on top of their respective groups and are guaranteed a home quarter-final.
The draw for the Intermediate Championship quarter-finals will be conducted on Monday morning, with second-placed finishers Westmeath, Monaghan, Roscommon and Tyrone also in the mix.
And in the Junior Championship, Carlow made it three wins from three in Group A, as Fermanagh opened their group series campaign with victory over Sligo.
In Group B, Louth overcame Limerick, as Longford defeated Kilkenny.
TG4 All-Ireland Senior Championship
Group 1 – Tipperary 2-4 Armagh 0-10
Group 2 – Kerry 4-13 Waterford 0-9
Group 3 – Cork 1-12 Galway 1-10
Group 4 – Mayo 2-15 Kildare 0-5
TG4 All-Ireland Intermediate Championship
Group 1 – Westmeath 2-12 Cavan 2-6
Group 2 – Monaghan 0-10 Offaly 0-10
Group 3 – Wexford 3-22 Antrim 0-4
Group 4 – Leitrim 0-17 Wicklow 0-10
TG4 All-Ireland Junior Championship
Group A
Carlow 1-11 Derry 3-4
Fermanagh 2-10 Sligo 3-6
Group B
Louth 0-14 Limerick 0-8
Longford 5-8 Kilkenny 2-6