Galway laid the foundation for victory in the opening half and led by 0-9 to 0-4 at the break.
However, Cork had to settle for a share of the spoils with Kildare on their return to Division 1.
Indeed, it took a late point from Katie Quirke to secure a draw on a scoreline of 1-10 to 2-7 for a Kildare side who led by 1-5 to 1-3 at the break.
Kerry began the defence of the league crown with an impressive 2-9 to 0-9 win over reigning All-Ireland champions Dublin in Clondalkin on Saturday.
Goals from Jadyn Lucey and Mary O’Connell were decisive for a Kerry side who led by 1-6 to 0-6 at the break in a game played in heavy rain.
Waterford, with goals from Chloe Fennell and Treasa NÃ Chrontaigh, secured an impressive 2-14 to 0-8 win in Armagh, having led by 1-5 to 0-3 at the interval.
In Division 2, a goal early in the game from Catherine Dolan helped Cavan lead Mayo by 1-8 to 0-8 at the break, but a goal from Sinead Walsh secured a draw in a contest which finished 1-13 apiece.
Cara McNamee and Niamh O’Neill scored two goals each as Tyrone notched a 7-11 to 0-11 win over Monaghan, while second half goals from Suzanne White and Cait Gillespie helped Donegal to a 2-11 to 1-6 win over a Westmeath side for whome Anna Jones converted a penalty in the opening half.
Tipperary, thanks to goals from Lauren Cunningham and a penalty from Emma Morrisey edged out Wexford by 2-12 to 2-9. Ciara Roban and Grace Donovan found the net for a Wexford side who trailed by 1-7 to 1-4 at the break.
Lidl National Football League Division One Round One
Kerry 2-9 Dublin 0-9
Cork 1-10 Kildare 2-7
Galway 0-16 Meath 0-8
Waterford 2-14 Armagh 0-8
Lidl National Football League Division Two Round One
Cavan 1-13 Mayo 1-13
Tyrone 7-11 Monaghan 0-11
Donegal 2-11 Westmeath 1-6
Tipperary 2-12 Wexford 2-9
Lidl National Football League Division Three Round One
Roscommon 3-6 Limerick 0-8
Laois 3-13 Fermanagh 3-13
Antrim 3-15 Clare 1-8
Louth 3-6 Down 1-12
Lidl National Football League Division Four Round Two
Leitrim 1-11 Offaly 1-8
Longford 3-8 Wicklow 0-7
Carlow 4-28 Kilkenny 0-1
Sligo 1-20 Derry 2-8