Though the visitors struck the woodwork twice in the first period, their menace diminished as United grew in stability in the second period - even if a spate of missed chances made for an uncomfortable end to the game.
"Galatasaray have got more experience in their team and in the first half we gave the ball away and they counter-attacked well from that," Sir Alex told Sky Sports. "I think [there was a] lack of concentration at times. It's unusual for us to be as frivolous with the ball.
"In the second half we did much better in that respect. I was pleased with the second half - it was more solid, we had better concentration and we saw it out really well.
"They never made any chances and all the chances fell to us - we could have scored four or five in the second half. They were very confident in their play and possession. You get that with these European teams.
"It's not a problem; you just have to be patient and make sure that when you're in possession you make it count and with missing the chances we did in the second half, we keep ourselves on the edge."
Nevertheless, Michael Carrick's early strike proved sufficient to get United off to a winning start in Group H, though the early pace-setters are Romania's CFR Cluj after a surprise victory over Braga in Portugal.
That result tees up an intriguing tie in Transylvania next month between the top two, though Sir Alex was just relived to see his side get off to a better start than in last term's ill-fated campaign.