Former Norway international Valon Berisha – one of six players cleared to play just hours before kick-off by authorities – netted their equaliser from the penalty spot after 60 minutes.
Paulus Arajuuri had given Finland the lead in the Group I encounter but Kosovo, playing their first competitive international, were impressive and had hit the bar through Leart Paqarada before Berisha earned a point.
It was one of three 1-1 draws in the group, with the clashes between Croatia and Turkey in Zagreb and Ukraine and Iceland in Kiev also ending level.
Croatia, playing behind closed doors, were pegged back by Hakan Calhanoglu late in the first half, just moments after taking the lead through an Ivan Rakitic penalty. The hosts had earlier hit the woodwork twice.
Ukraine’s Yevhen Konoplyanka hit the post with an 83rd-minute penalty after team-mate Andriy Yarmolenko had cancelled out Alfred Finnbogason’s early opener for Iceland.
Spain hit form as they opened with an 8-0 thrashing of Liechtenstein in Leon in Julen Lopetegui’s first competitive match in charge.
Diego Costa, David Silva and Alvaro Morata all struck twice with Sergi Roberto and Vitolo also on the scoresheet. Seven of Spain’s goals came in a rampant second-half display.
That made them the early leaders of Group G but Italy also opened with a win despite being reduced to 10 men against Israel in Haifa.
Graziano Pelle got Giampiero Ventura’s men off to a fine start before Antonio Candreva added a second from the penalty spot. Tal Ben Haim pulled one back with a fine chip but Ciro Immobile sealed a 3-1 win late on despite the sending-off of Giorgio Chiellini.
The other match in the group was not completed as Albania’s clash with neighbours Macedonia in Shkoder was abandoned after 77 minutes due to torrential rain.
The score was 1-1 at the time after Ergjan Alioski replied to Armando Sadiku’s early effort.
Gareth Bale scored twice as surprise Euro 2016 semi-finalists Wales began their campaign with a 4-0 defeat of Moldova in Group G.
The Real Madrid star wrapped up victory with an injury-time penalty after scoring his side’s third early in the second half at the Cardiff City Stadium. Sam Vokes and Joe Allen scored before the break.
In another game that almost fell victim to severe weather, substitute Daryl Murphy headed an 81st-minute equaliser as the Republic of Ireland claimed a 2-2 draw with Serbia in Belgrade.
Jeff Hendrick had given the Irish a third-minute lead but Serbia responded through Filip Kostic and with a Dusan Tadic penalty.
Goals from Martin Hinteregger and Marc Janko helped Austria to a 2-1 win over Georgia in Tbilisi. Valeri Kaziashvili hit the woodwork for Georgia, who pulled one back through Jano Ananidze but Austria held on.