Feeling Defeated

My son has been EBF and is 3weeks old. Today he nursed for 3 hours straight, taking maybe a 10 min break here and there because I thought he was done because he would fall asleep or fall off and not want to relatch. I'd lay him down and he wouldn't make it long before crying and rooting again. I kept switching sides and trying to get him full. Finally I gave in and gave him 2oz of expressed breastmilk in a bottle. I'd been trying to make it 6 weeks before I did this (he will need to take a bottle before I go back to work at 12weeks), but I feel like I failed. He seems content now that he ate the bottle. I'm just worried because I don't have much milk saved up yet and I wanted to keep breastfeeding a little longer. Everything has been going well and we have survived a cluster feed before but this was different which is why I gave in. I'm drinking a lot of water and going right back to the boob once he wakes from this nap.