Scotland could have taken the lead as early as the first minute when Mark Burchill knocked the ball wide of the post after a fine cross from David Graham. Burchill continued to torment the opposition but his match came to a premature end after an accidental collision with team-mate Barry Elliot six minutes from the end saw him carried off.
"He has a slight concussion," Tommy Craig, the Scotland Under-21 manager, said. "He appears to be okay but we'll take him to a local hospital for X-rays as a precaution."
Burchill could have had a hat-trick in the first half, but headed straight into the Lithuanian goalkeeper's arms from close range in the 35th minute. Then, three minutes later, Burchill fired the ball over the bar after a knock-down from Graham had put him in the clear.
Lithuania had plenty of possession but their shooting was erratic, and Alexander's first real test of action came on the stroke of half-time. Arturas Fomenko headed into the box, but the Scotland No 1 was alive to the danger and took it off the head of Tadas Graziunas.
Alexander again had to look lively four minutes after the break when he parried a shot from Vidas Kauspadas, then recovered to pounce on the rebound as Fomenko raced in.
Scotland were beginning to wilt under the weight of attacks and Alexander could only watch helplessly in the 70th minute as a 25-yard shot from Tomas Danilevicius rattled his post. A shot by Burchill, blocked by the keeper, momentarily raised the siege, but Alexander was soon in action again.
Three minutes from time he pushed a shot from substitute Aivaras Laurisas around the post and then saw an effort from Fomenko come back off the bar. In injury-time Alexander made another great stop to defy Danilevicius as he raced clear from the centre circle.
After such a poor qualifying campaign last time round, Craig was encouraged by his side's start: "We deserved a point. I thought we were well disciplined and all the players put in a tremendous shift."
Lithuania Under-21: Leus, Joksas, Gleveckas, Zutautas, Dziaukstas, Gardzijauskas, Cikas, Kauspadas, Danilevicius, Fomenko, Graziunas.
Scotland Under-21: Alexander, McEwan, Buchan, McCluskey, Archibald, Brebner, Easton, Strachan, Burchill, Graham, Campbell.
Referee: G Buzas (Latvia).