When 39-year-old Giggs replaced Marouane Fellaini after 66 minutes at the raucous Donbass Arena he made his 145th appearance in the competition, including qualifiers, to overtake Spaniard Raul.
The Welshman, who is also now part of United's coaching staff, won the competition with United in 1998-99 and 2007-08 and also has 13 league titles to his name.
He has made over 900 appearances for the club.
"It's fantastic," Moyes said. "I was telling him before the game he was going to come on off the bench.
"What a record. What a player," the Scot added, but he will be disappointed with the result after his side had looked in control of a match they led until the 76th minute.
United, missing the injured Wayne Rooney but with Robin van Persie back in attack, went ahead after 18 minutes when Fellaini pulled the ball back for Danny Welbeck to stab home from close range after defender Jaroslav Rakitskiy failed to clear.
Shakhtar had plenty of possession but came up against the brick wall of a United defence until 14 minutes from the end when Rakitskiy made amends.
The result gives United and Shakhtar four points each at the top of Group A. Bayer Leverkusen are third with three points after a 2-1 home win over Real Sociedad.
Moyes, not surprisingly, looked for the positives but also lamented United's habit of giving the ball away, particularly in the second half as Shakhtar pressed for an equaliser.
"It is a good point. They have a great record against English teams and the team stood up well to it. We got a good goal early doors, and I thought we were going to stop them scoring," Moyes said.
"We said at half time we needed to keep the ball better, but we actually gave it away more in the second half so I'm a bit disappointed with that. But we were really unlucky to lose a goal."
Goalscorer Danny Welbeck added: "Four points from the first two games is okay, considering we were away at Shakhtar. We are a bit disappointed as we are Manchester United and want to win every game, but we have to be satisfied with a point."
"Both teams should be happy with the result. Manchester United went back after scoring early and played on counter-attacks. They are great at this," Shakhtar's Romanian coach Mircea Lucescu.
"But in general we controlled the game and did not allow them much. It was agreed to force the tempo in the final 20 minutes and we really stuck to our plan."