Kieran Donaghy hit Kerry's goal early on as Galway, in contrast, missed several goal chances.
There was a breathless start to the second quarter-final as Fintan Cregg and Ciaran Murtagh hit Roscommon goals before Lee Keegan netted for Mayo.
As the pace slowed, both sides missed a host of chances in a nervy second half.
After Paddy Durcan kicked Mayo ahead in the 68th minute, Cillian O'Connor missed a couple of chances to extend their lead before Donie Smith levelled with a free from over 45 metres.
O'Connor then had another long-range chance to snatch victory for Mayo but left the effort just short.
The game started in dramatic fashion as Cregg lobbed a superb finish over David Clarke before Murtagh blasted a second goal two minutes later.
However, within a minute Keegan charged forward to net for Mayo, although his shot took a wicked deflection off Niall McInerney.
As Roscommon lost their way, Mayo moved 1-8 to 2-3 ahead by half-time and at that stage, Stephen Rochford's side looked the likely winners.
But the half-time break came at the right time for Roscommon as they regrouped to hit three straight points after the resumption.
Both sides seemed produced some desperate finishing during the remainder of the contest with Mayo particularly guilty and that meant that a draw was probably the appropriate outcome.
In Sunday's first game, Galway missed a series of goal chances, with Kerry goalkeeper Brian Kelly making a couple of great saves.
Kelly denied Ian Burke early on before Donaghy hammered in Kerry's goal after soaring to win possession.
Kerry led 1-10 to 0-8 at half-time and held on as Galway missed three more goal chances in the second period.
Damien Comer had a goal opportunity blocked immediately after the restart and Galway's challenge faded badly after Kelly denied Sean Armstrong shortly afterwards.
However, Kerry boss Eamonn Fitzmaurice is unlikely to be happy with his team's performance after Galway were able to create several goal opportunities.
The Kingdom, who have won a record 37 All-Ireland titles, are scheduled to face the winners of the Roscommon and Mayo tie on 20 August.
Kerry reached the All-Ireland Football semi-finals as they beat Galway 1-18 to 0-13 while Mayo and Roscommon must replay after a 1-12 to 2-9 draw.