Allan McGregor made a great save from Argentina skipper Javier Mascherano before Rodriguez finished off a fine move by sliding the ball past the Scotland keeper from close range.
The South Americans, slick and speedy at times, appeared to ease off after their blistering start but still dominated most of the match, leaving the Scots looking for their first friendly win at the national stadium since 1996.
Most of the attention before the game was on Maradona, who returned to Glasgow for the first time since he scored his first goal for Argentina in a 3-1 friendly win at Hampden in 1979.
The former Napoli and Barcelona superstar was given a huge roar of approval from the Tartan Army, mostly for his 'Hand of God' goal against England in the 1986 World Cup.
There was a minute's silence for former Scotland manager Ian McColl and Pedro Pompilio, the President of Boca Juniors and First Vice President of the Argentinian Football Association, who both passed away in recent weeks.
Maradona must have been pleased when the action started as the visitors produced an opening 10 minutes of breathtaking football after which they were ahead and cruising.
Defender Martin Demichelis headed a Joan Gutierrez corner just past the post after a minute.
Five minutes later, with the South Americans in complete control, McGregor was forced into a fabulous save from Mascherano's long-range volley.
Argentina scored the opener with another fine move that split open the Scotland defence.
The tireless Carlos Tevez picked up the ball on the right-hand side of the box, slipping it to Gutierrez, whose short, first-time pass put Rodriguez in and the Atletico Madrid player clinically slid the ball past McGregor from close range.
In the 19th minute, with no change in the pattern of play, Tevez sent Ezequiel Lavezzi through on goal with a slide-rule pass and McGregor had to block the angled shot from close range with his feet.
It looked as if it was going to be a long night for George Burley's side but at the other end moments later, with Scotland's first attempt on goal, James McFadden's shot from a Kris Commons' cross was blocked for a corner by Demichelis, which the visitors survived.
However, Argentina bared their teeth again in the 35th minute when defender Javier Zanetti raced down the right and cut the ball back to Tevez, who had time to take a touch on the edge of the box before blazing over the bar.
Then five minutes from the break, some hesitation from Demichelis inside the Argentina penalty area as he ushered the ball back to his keeper allowed McFadden to rob him but the Birmingham striker's low drive was too weak to bother Juan Pablo Carrizo.
Chris Iwelumo, ineffective in the first 45 minutes, was replaced by Aberdeen striker Lee Miller at the interval and he was given time to ease himself into the game as the visitors failed to replicate their opening salvo.
But in the 51st minute, after Scotland tried unsuccessfully to pass their way out of defence, Lavezzi drove over the bar from just inside the box.
It was indicative of the South Americans' more casual approach to the game, although there was still the occasional shaft of skill to light up proceedings, provided by Tevez in the 58th minute when he burst forward with pace only to once again drive over.
Argentina dropped the tempo and Scotland sensed there was a way back but there appeared to be a lack of cutting edge to the home side's play.
In the 70th minute, Miller passed up a great chance to equalise when he headed a corner from substitute Shaun Maloney, on for Paul Hartley, past the post from around six yards out.
Three minutes later, after McManus had fouled Demichelis 25 yards from goal, Gabriel Heinze curled a free-kick two yards over the bar.
The game petered out and the Scottish fans in the 32,492 crowd left the ground perhaps grateful the visitors had not registered a more convincing win.