Are y'all kidding me? If we're talking a real Mongol army, not something like the glorified scouting expedition the Mamelukes fought to a bloody draw at Ain Jalut, but a real killing machine commanded by the Great Khan himself, or Subedei or Jebe like the armies which ensured that the Persian areas of central and southwestern Asia did not exceed the GDP they had in 1215 CE until the 1950's, against a Roman horde commanded by Marius, Caesar, or Agrippa (the best Roman commander of the era would have to be one of those three, right?), and the Mongols would utterly destroy them, ten times out of ten. All of you prattling on about it mattering what sort of terrain the fight takes place on are ignoring one very simple difference between the two: the Mongols are a cavalry force with the capacity to fight a siege and a tradition of excellent scouting and spycraft, whereas the Romans are pretty much just infantry and basic artillery (given that their Numidian auxilliary cavalry would be chased away in about 16 seconds) with next to no tradition of scouting whatsoever. Between the superior mobility of the Mongols, their advance knowledge of pretty much any terrain upon which they fought (the exception again being Ain Jalut) and the fact that the Romans could not maintain any sort of logistical tail (anything lagging a day or more behind the main Roman army would just wind up feeding and supplying the Mongols, considering who easily and regularly outider detachments would wind up looting it), there's no way the battle would be fought on any grounds not of the Mongols' choosing, unless they were trying to capture Rome itself and the battle was fought in the Italian peninsula itself, within a day's march of Rome or Mediolanum. There'd be a brief battle, while the Mongol achers on their tough steppe ponies turned the legions first into tortoises, then pincushions, and then there'd be about a two or three day trail of dead bodies heading off in whatever direction the Mongols chose to allow the routed Romans to retreat so as to utterly break their army, and a small handful of legionaries would trickle back home to spread news of the catastrophic defeat and awesome nature of their foes.
The interesting question, to me, is not whether the legions could defeat the Mongols (they couldn't), but whether Tilly's tercios could. Now, one of the advanced Thirty Years' War armies, the Empire under Wallenstein, the Swedes under Gustavus, Torstenssen, or Oxenstierna, or the Spanish under the Cardinal-Infante, would have enough firepower and mobility to deal with the Mongols, provided they could live off the land, but a slow moving tercio might not be able to deliver enough of a punch to the Mongols to knock them out.