The Young Dragons, drawn alongside England in Group One, had started their qualification bid with a 1-0 win over Moldova but were thumped 5-1 at home by Finland last time out.
The trip to San Marino offered a chance to respond but a solitary goal by Juri Biordi was sufficient for the hosts to claim victory.
Geraint Williams' side made a positive start, forcing a corner in the first two minutes before Colombini had a shot but there was little in the way of clear-cut chances.
Wesley Burns missed the target for Wales but the game was a bit stop-start until midway through the first half when San Marino took the lead.
Manuel Battistini provided the assist and Biordi converted.
Burns had a shot blocked as the visitors attempted to hit back but they did not really threaten to break through. Instead, San Marino pressed but Marco Bernardi failed to find the target.
The Welsh had a brief flurry before half-time but Thomas Lawrence and Jake Cassidy could not make it count.
Bradshaw then finally forced San Marino goalkeeper Elia Benedettini into a save with a little over 10 minutes remaining.
With time running out, Wales desperately sought an equaliser, Lawrence's effort on goal repelled by Benedettini.
Bradshaw had a shot blocked and with just two minutes remaining Wales missed a good opportunity to equalise, Robert Ogleby missing the target.