The visitors had the better of the opening 45 minutes, only for Robert Beric to give the hosts the lead three minutes before half-time.
Gylfi Sigurdsson had a sight of goal after only two minutes, but the Icelandic international could only fire wide.
Despite enjoying the majority of possession, Maribor failed to carve out any meaningful chances in front of goal.
Vertonghen should have done better when his header from Tom Huddlestone's cross was easily saved, while Sandro lashed the ball narrowly wide from the edge of the area.
However, it was the hosts who made the breakthrough in the 42nd minute when Beric slotted in from close range.