Scotland's double cup winners held their own against the European heavyweights and took a shock lead from the penalty spot on 58 minutes when goalkeeper Fernando Muslera pulled down Chris Kane in the box to receive a straight red card.
Club captain Jason Kerr slotted his spot kick past replacement goalkeeper Berk Balaban to silence the home crowd inside the Basaksehir Fatih Terim Stadium, but the lead lasted only two minutes as Sacha Boey levelled the tie.
Saints could not make the most of their man advantage but Stevie May had a shot cleared off the line as Davidson's men set up a huge night in Perth next Thursday, with a place in the play-offs up for grabs.
"I'm obviously delighted," Davidson told the BBC. "You come to a place like Istanbul and get a result against a fantastic Galatasaray, it's a phenomenal result for us."We had to work really hard and we tried to do things right.
"Galatasary had the majority of the possession and created chances, but I think we managed to create a few chances ourselves and obviously the sending off and the penalty was a big turning point in the game.
"It gives us hope to go into the second leg with something to play for.
"We know it's just the first game and Galatasaray are a very good team and they'll come at us. They've got a lot of talented players so we need to make sure we're ready and give ourselves a chance to try and get a result and make it difficult for them."
Saints survived an early onslaught at the stadium which is named after Galatasaray's manager but belongs to one of their Istanbul rivals, Basaksehir, with their own home under renovation.
Mostafa Mohamed missed a sitter three minutes in when he volleyed over from six yards and Kerem Akturkoglu hit the post from 15 yards five minutes later.
Saints settled and Murray Davidson and David Wotherspoon had efforts saved, with the latter having a particularly good chance only to shoot straight at the goalkeeper.
The hosts also started the second half strongly but Saints were handed a gift when a poor back-pass put home 'keeper Fernando Muslera in trouble and he pulled down Saints striker Kane to concede a penalty and receive his marching orders.
After a long delay, centre-back Kerr held his nerve to convert the spot-kick.
Saints could not enjoy their lead for long. On-loan Manchester United left-back Reece Devine tried to cut out a pass to French right-back Boey but missed the ball, and the debutant - who had been linked with Celtic this summer - took advantage, drilling an angled drive into the far corner.
Balaban almost gifted Saints another goal, losing the ball to May near the corner flag, but the substitute's effort from a difficult angle was cleared off the line.The 10 men finished strongly. Jamie McCart produced a great headed block from Marcao's volley before Clark made a brilliant save from Christian Luyindama's header to secure a famous result for the Perth men.