Spoljaric's penalty four minutes from time vaulted the Cypriots into second place behind Spain on 12 points, but they still have two difficult matches to come, away to the group leaders and Austria.
If they did make the play-offs, the Cypriots could find themselves in the draw with England after Sheffield Wednesday's Niclas Alexandersson came off the bench to score the goal that gave Sweden a 1-0 win over Bulgaria on Saturday that left the Swedes almost certain to win Group Five.
In Scotland's section, the runaway winners the Czech Republic won their eighth consecutive group game, 4-0 against Lithuania with two goals each from Pavel Nedved and Jan Koller. Estonia - Scotland's opponents on Wednesday - remained in the hunt with two goals in the last two minutes against the Faroe Islands.
Jan Dahl Tomasson's 82nd-minute winner for Denmark against Switzerland hampered Wales' hopes of making the play-offs from Group One.
France face a struggle to qualify from Group Four after drawing 0-0 with Ukraine in Kiev.