Robert Lewandowski got Poland out of jail as his injury-time goal sealed a 2-1 win over Armenia.
Armenia looked to have battled to a creditable draw, having played since the 30th minute a man down following Gael Andonian's two yellow cards, and subsequent red.
Lewandowski first pounced just after half-time but the visitors equalised through Marcos Pizzelli's free-kick shortly afterwards.
The Bayern Munich striker then settled the game in the fourth minute of stoppage time when he got his head to a Jakub Blaszczykowski cross and Poland joined Montenegro on seven points at the top of Group E.
Fatos Beciraj was Montenegro's match-winner against Denmark in Copenhagen.
The Dynamo Moscow forward struck the only goal of the game, while the third match in the group saw Romania draw 0-0 in Kazakhstan.
A pair of goals in four first-half minutes saw Germany coast to a 2-0 victory over Northern Ireland in Hanover.
Julian Draxler's opener and Sami Khedira's header both came inside the first 17 minutes as the reigning world champions maintained their 100%.
The other games in Group C saw Czech Republic and Azerbaijan draw 0-0, while Norway got their campaign up and running with a 4-1 success over San Marino.
Norway led through Davide Simoncini's 11th-minute own goal but San Marino equalised nine minutes after half-time through Mattia Stefanelli.
Adama Diomande restored Norway's lead with a curling effort from the edge of the penalty area in the 77th minute before Martin Samuelsen's powerful 12-yard effort gave Per-Mathias Hogmo's team a cushion.
Bournemouth forward Josh King completed the scoring and made it a more respectable scoreline in Norway's favour with an 83rd-minute finish.
England remain top of Group F despite a goalless stalemate in Slovenia, while Lithuania are up to second spot after late goals from Fedor Cernych and Arvydas Novikovas, from the penalty spot after Jonathan Caruana was sent off for Malta, gave them a 2-0 victory.
Scotland lost 3-0 in Slovakia after a Robert Mak double and another strike from Adam Nemec eased the hosts to success.