Vladimir Petkovic's visitors classily controlled long periods of the contest but Northern Ireland's defence was expertly marshalled by Jonny Evans and the veteran Gareth McAuley and held firm until referee Ovidiu Hategan intervened in the 57th minute.
Corry Evans blocked a thunderous strike at close quarters from Xherdan Shaqiri and the Romanian official's decision to give a penalty for handball was as rushed as it was ill-advised.
AC Milan fullback Rodriguez, a threat throughout the match through his fine deliveries from the left, made no mistake from 12 yards to leave Switzerland in the driving seat ahead of Sunday afternoon's (Monday morning AEDT) return in Basle.
Michael O'Neill's Northern Ireland are faced with considerable odds to overturn and must hope one more flicker of the magic that took them to EURO 2016 and to the brink of a first FIFA World Cup since 1986 remains.
O'Neill preferred Kyle Lafferty to Conor Washington in attack and the well-travelled Heart of Midlothian striker was lively during the opening exchanges.
Switzerland centre-back Fabian Schar hacked a testing Josh Magennis cross out of his own six-yard box before earning a fifth-minute booking for scything through Stuart Dallas.
The visitors settled well amid a raucous Windsor Park atmosphere, with Arsenal midfielder Granit Xhaka twice going close via drives from the edge of the penalty area before McAuley made a similarly testing clearance to Schar's earlier effort.
West Bromwich Albion defender McAuley was caught out underneath a lofted pass from Shaqiri in the 18th minute but Michael McGovern saved superbly down to his left to keep out Haris Seferovic's cushioned volley.
Seferovic was unable to direct a cross from captain Stephan Lichtsteiner goalwards 10 minutes before the break and Northern Ireland were able to reach halfway on terms.
Within 30 seconds of the restart, Shaqiri came agonisingly close to breaking the deadlock – collecting Rodriguez's raking cross and whipping in a rising shot that clipped the top of the crossbar.
A stretching Seferovic was unable to apply the finishing touch with the goal at his mercy in the 54th minute but Rodriguez made no mistake when he was erroneously given the chance to open the scoring shortly afterwards.
Corry Evans turned away and blocked Shaqiri's rasping volley just inside the Northern Ireland area – the ball striking his back just underneath his right upper arm. Hategan's positioning was far from ideal but he blew quickly to point unfairly towards the spot.
With Corry Evans booked for his supposed offence – a caution that rules the Blackburn Rovers man out of the return leg – Rodriguez kept his cool amid a chorus boos to send McGovern the wrong way.
Shaqiri spurned a chance to compound Northern Ireland's misery when McGovern snaffled his botched cutback before the hosts wasted a rare chance with 18 minutes remaining.
Chris Brunt's set-piece delivery was excellent but Magennis could not find the finish to match and his header flew wide.
A fairly fanciful penalty shout for handball against matchwinner Rodriguez was the most notable attacking moment for Northern Ireland during the closing minutes. Given Hategan's earlier intervention, it felt like a question they were entitled to ask.