The result left Denmark second on 19 points from nine games, three behind leaders Poland and three ahead of third-placed Montenegro, who now have only a slim chance of finishing as the group's runners-up.
Denmark are at home to Romania in their final game on Sunday afternoon (Monday morning AEDT) while Poland, who thrashed Armenia 6-1 away, are at home to Montenegro knowing that they will go through with a draw.
Montenegro made the brighter start in the cauldron of their Podgorica stadium as they pressed forward in numbers but were stung by a moment of individual brilliance by Eriksen.
The Tottenham Hotspur playmaker gave goalkeeper Danijel Petkovic no chance with a rasping shot from 20 metres with his weaker left foot in the 15th minute, after Henrik Dalsgaard headed a long ball into his path.
The home side suffered another blow five minutes later after their captain and top scorer Stevan Jovetic limped off with a thigh injury, leaving Montenegro's three-pronged attack with no cutting edge.
Denmark winger Pione Sisto tormented his markers throughout and forced a good save from Petkovic with a ferocious drive in the 65th minute as Montenegro struggled to make any impact at the other end.
Eriksen and substitute Nicklas Bendtner missed sitters in the closing stages as Montenegro threw caution to the wind but had nothing to show for their industrious approach.
Europe's nine group winners will qualify automatically for next year's 32-team tournament in Russia while the eight best runners-up will enter two-leg play-offs for the remaining four berths.
