The game - Ton's second in 24 hours after Wednesday's 10-2 win at Campbeltown Former Pupils - was a tight affair in which the hosts gave a good account of themselves.
Despite being on the receiving end of something of a going-over from Concordia's compatriots Rapid Bucharest recently, Morton were only undone when a lapse in concentration gifted Silviu Pana the winner in the second half.
With Concordia promising to provide a stern test after recently beating St Johnstone 4-0 and losing 3-2 to Celtic, Ton chief Allan Moore took the opportunity to run the rule over a handful of experienced trialists.
Former Queen of the South right-back Craig Reid and ex-Dunfermline centre-back Kevin Rutkiewicz featured in a strong back four which also included summer signings Willie Dyer and Mark McLaughlin, who was making his first appearance for the club.
In midfield, former Falkirk, Bradford City and Scotland Under-21 cap Chris Mitchell played on the right with Martin Hardie inside him.
There was a familiar look to the rest of the line-up. Michael Tidser and David O'Brien made up the left-hand side of the midfield, while Peter Weatherson and Archie Campbell were paired up front.
Of the side that played at Kintyre Park the previous evening, only Tidser, Gaston and Weatherson started again.
Ton began brightly and two of the aforementioned trialists combined as the hosts went close to opening the scoring in the 11th minute.
Mitchell sent over a deep corner from out on the Cowshed side which imposing defender Rutkiewicz met with a looping header that crashed back off the junction of bar and post.
For their part, Concordia, who finished ninth in Liga I last season - their second ever at the top level after spending most of their history in the third tier - played some sharp passing football.
However, their approach play lacked a cutting edge in the final third - although the resolute defending of a well-organised Ton rearguard was also instrumental in limiting their chances.
In the first half, they were restricted to two shots from distance in open play, one of which was drilled into the side netting by Sorin Paraschiv, eliciting cheers from the Romanian contingent who mistakenly thought the ball had nestled in the back of the net.
The other effort, their best of the first 45, was a fine long-range strike from left-sided midfielder Florin Purece, but Derek Gaston was alert and sprung to his left to parry.
Concordia should, however, have taken the lead in the 32rd minute when former Romania international Paraschiv's inswinging corner found Valentin Simion lurking unmarked four yards out.
But the 25-year-old frontman flicked a weak effort against the outside of the right-hand post and wide for a goal kick.
At half-time Moore replaced Reid, Rutkiewicz and Weatherson with Ewan McNeil, St Mirren loanee Jason Naismith and Stephen Stirling.
Meanwhile, Hardie was pushed forward to partner Campbell, a move which Moore later revealed he would consider utilising next season.
Morton again started quickest, with McLaughlin demonstrating his aerial prowess three minutes into the second half by outjumping a static defence to head a Tidser corner narrowly over the top.
His side's continued failure to break down Ton's stubborn resistance was beginning to get to Concordia head coach Ilie Stan, who threw his baseball cap to the ground in a fit of anger after watching another ambitious pass fail to hit its intended target.
His rage soon subsided though when substitute Pana scored to give the Romanians the lead in the 71st minute.
Gaston passed short to Naismith, who looked to bring the ball out from the back but turned straight into Concordia striker Welington Carlos Da Silva.
The Brazilian trialist darted to the bye-line and rolled the ball across the face of goal and past Gaston for the inrushing Pana to tap into an empty net.
Second-half substitute Jordan Halsman went close to restoring parity in the 84th minute with a low drive which forced a fine reflex save from Estonian keeper Sergei Lepmets. Back at the other end, Gaston made a stunning stop of his own two minutes later when he tipped a thunderous Paraschiv drive onto the face of the crossbar.
The save was the last piece of notable action in an encouraging 90 minutes in which a strong Morton team competed well with the Romanian top-flight side.
First Half (4-4-2): Gaston; Reid [Trialist], Rutkiewicz [Trialist], M. McLaughlin, Dyer; Mitchell [Trialist], Hardie [Trialist], Tidser, O'Brien; Weatherson, Campbell.
Second Half (4-4-2): Gaston; McNeil [Trialist], Naismith, McLaughlin (O'Ware, 55), Dyer; Mitchell (Wallace, 59), Stirling, Tidser (Wilkie [Trialist], 55), O'Brien (Halsman, 59); Campbell, Hardie (Hawke, 67).
Sub not used: Flack (gk).
(4-4-2): Lepmets; Ghionea, Belu, Maxim, Belica; Paraschiv, Popa, Rocha, Purece; Welington, Simion.
Subs: Cretu, Leca, Damian, Muzac, Pana, Ciupitu.