Favourites Croatia were held 1-1 at home by Slovenia in another surprise result for the underdogs, but Spain restored some order with a 2-1 victory at home to Norway.
The five playoff winners will join the other 11 known finalists in the European Championship being held in Portugal from June 12-July 4 next year. All the second legs will take place next Wednesday.
The first goal of the day came in the 200th international match played at Hampden Park, which has just celebrated its 100th anniversary.
Winger James McFadden fired the winner before 51,000 delighted home fans as Scotland earned their first competitive win over the Dutch since the 1978 World Cup finals.
The Netherlands had an outstanding team on paper with Ruud van Nistelrooy and Patrick Kluivert up front, supported by Edgar Davids and Andy Van der Meyde in the middle.
But as in 2001, when they were surprisingly dumped out of the World Cup playoffs by Ireland, the Dutch failed to combine effectively as a team and squandered their scoring chances.
"It was a great performance in the first half, we played brilliant football," said Scotland coach Berti Vogts.
"In the second, the legs were a little bit tired but it was organised and we functioned as a team."
Disappointed Dutch skipper Frank de Boer said: "This is a terrible result for us."
"Now we have to score at least two goals in Amsterdam on Wednesday, and that may be difficult considering the number of chances we missed today."
Wales, without two of their most influential players after long-term injuries to midfielder Simon Davies and forward Craig Bellamy, faced sustained pressure in Moscow.
But despite being dominated for much of the game, Wales take their qualifying hopes back to Cardiff, while Russia will be without suspended playmaker Alexander Mostovoi and goalkeeper Sergei Ovchinnikov after both were booked on Saturday.
Despite encouraging results, Scotland and Wales will have their work cut out to join England, who qualified automatically as group winners, to complete a British trio in Portugal.
Britain last had three teams at a major championship in 1986,
Latvia joined the surprise packages by beating a Turkey side reduced to 10 men after a 73rd minute red card for defender Emre Asik.
The Latvians, ranked 69th in the world, secured victory over a side who have struggled since finishing third at the 2002 World Cup. A 29th minute strike by Maris Verpakovskis, his fifth of the qualifying campaign, proved to be the winner.
Though Senol Gunes's men will still expect to qualify, the year is clearly turning sour after his former Group Seven leaders were pipped to the top slot by England.
Along with Asik, Turkey will also be without keeper Rustu Recber and defender Fatih Akyel next week after yellow cards on Saturday, while Latvia will miss defender Valentins Lobanovs.
Spain, surprisingly forced into the playoffs after finishing second to Greece, suffered an early shock in Valencia when they fell behind to Steffen Iversen's 14th minute opener for Norway.
Raul equalised after 21 minutes and Spain's overwhelming second-half pressure eventually paid off when Henning Berg turned the ball into his own net under pressure from Raul five minutes from time.
Goal-happy Dado Prso, who 10 days ago scored four times in Monaco's sensational 8-3 Champions League win over Deportivo Coruna, gave Croatia a fifth minute lead in Zagreb.
But Ermin Siljak levelled for Slovenia midway through the first half to swing the match their way -- although a later booking means he will miss the second leg.
With the away goals rule in operation, a goalless draw next week will give Slovenia a remarkable third major tournament appearance in four years after Euro 2000 and the 2002 World Cup.
Euro 2004 hosts Portugal and European champions France were also in action on Saturday as they warmed up for the tournament with friendlies against Greece and Germany respectively.
The Portuguese were held to a 1-1 draw in Aveiro after Luis Figo missed a first-half penalty, while the French sauntered to a 3-0 win over rivals Germany in Gelsenkirchen and a record of 13 wins in a row to eclipse the previous mark set in 1984.