Albion earn promotion to the third tier while East Fife, who held a 2-1 lead after the first leg, are relegated to League Two.
In the first half both sides had chances, but neither could break the deadlock, with Pat Clarke going closest for the hosts.
Stirling took the lead in the 73rd minute when White beat Dylan Rooney from a tight angle.
Nathan Austin then missed a great chance to put East Fife back in front on aggregate before Stirling went ahead in the tie through Cunningham's powerful finish.
Liam Buchanan went close to levelling in the dying moments, but the hosts couldn't find a way through and will now be playing in the fourth tier next season.