Wales took the lead at Stadion Antona Malatinskeho through Moore's maiden goal 25 minutes into his first competitive start for the senior side.
But they were denied a third successive victory by Juraj Kucka's goal early in the second half as Slovakia, who had Norbert Gyomber sent off late on, fought back strongly for a point.
The point leaves the visitors fourth in Group E and three points behind Slovakia, although they have a game in hand on each of the sides above them.
Wales were 1-0 winners when these sides met in Cardiff seven months ago and they took a deserved lead in Trnava.
Less than a minute after Gareth Bale's looping header struck the angle of post and crossbar, Moore got on the end of Daniel James' cross from the left to head into the bottom-left corner.
Robert Mak squandered a good opportunity to level before half-time when firing straight at Wayne Hennessey from a promising position, but Slovakia were on level terms eight minutes into the second half.
Connor Roberts' headed clearance fell nicely for Kucka, who rifled a volley past Hennessey from the edge of the 18-yard box.
Hennessey was called into action to keep out Albert Rusnak's low drive as Slovakia searched for a winner, but Kucka was shown a second yellow for clattering into Moore two minutes from time to end their hopes of finding one.
What does it mean? Wales' qualifying hopes still in the balance
Wales may be down in fourth place in Group E after this draw, but a point is far from disastrous.
The Dragons' qualifying hopes are still ultimately in their own hands if they can win their remaining three matches.
Giggs' gamble pays off
Wales boss Giggs made a huge call in selecting Tom Lockyer and Moore for their full competitive debuts in defence and attack respectively.
Both players justified their selection, particularly in the case of Wigan Athletic striker Moore as he marked the occasion with an impressive headed goal.
Davies sloppy in possession
Ben Davies has endured a torrid time of things at club level with Tottenham this season and he was far from his best against Slovakia.
Recalled from the 2-1 win over Azerbaijan, the left-back lost possession 16 times - only James, tasked with taking players on, did so more often in a Wales shirt.
Key Opta stats
- Wales lost 5-1 in their first meeting with Slovakia in October 2006 but have gone unbeaten against them in their four matches since (W3 D1).
- Slovakia have won just one of their last five European qualifiers on home soil: a 2-0 victory against Hungary in March. They have drawn two and lost two.
- Wales have taken seven points from their opening five qualifiers for Euro 2020 (W2 D1 L2) – their lowest points tally at this stage of a European Championship qualifying campaign since the doomed effort to reach Euro 2012 when they took just three points from their first five games.
- Kieffer Moore netted his first goal for Wales. It was also his first goal in any competition since netting for Barnsley against Bristol Rovers in League One in May. Moore left Barnsley to join Wigan in August.