Caesar and Pompey's forces battled with one another on numerous occasions during the Fall of the Roman Republic. What is the earliest battle that is commonly accepted as the one where it became clear that Caesar's eventual victory was inevitable?
When Caesar led his army across the Rubicon River, Pompey fled to Greece with his army and most of the Senate.
In the beginning, the advantage seemed to lie with Pompey. Not only did he command a large army, but he also controlled much of the East as well as Spain. Caesar, however, handled the situation masterfully. Finally in 48 B.C., Caesar crushed Pompey's army at Pharsalus in Greece.
Although Caesar's crossing of the Rubicon, which marked the border of his province, was significant in that it marked the start of full on hostilities, it was by no means certain that he would be victorious. According to both Plutarch and Suetonius, Caesar had doubts in his mind as he reached the river, the latter claiming that these doubts were soothed only by the appearance of a divine apparition. Appian also wrote that Caesar had concerns, although those had more to do with the bloodshed that he might unleash, while Cassius Dio does not mention any hesitation.
Misinformation about the size of Caesar's army prompted Pompeius and many of the senators to leave Rome not long afterwards. According to Appian and Plutarch, Pompeius was forced into this action by others in spite of his military experience. Cassius Dio attributes the move to Pompeius fearing that the masses in Rome would support Caesar over him. In terms of numbers, Pompeius had a superior force and recruited further troops in the countryside. It was around this time that Labienus, Caesar's long time deputy in Gaul, switched his allegiance to Pompeius. Even if he had grown distant from Caesar and was afraid of being harmed, as Cassius Dio records, such a move would have been extremely foolish if Pompeius was already looking like a lost cause.
After consolidating power in Italy and Spain, Caesar crossed the Adriatic Sea. However, his supply lines were cut and his defeat at Dyrrachium could well have ended the war in favour of Pompeius. That this did not occur was because the Pompeius was overly cautious, and according to Caesar, fearful of an ambush, having far exceeded his expectations in the fight.
It was only at the final battle at Pharsalus that matters were decided, and even there, Caesar was in a far from solid position. He was short on supplies, only able to acquire them by forage. According to Caesar, such was the confidence in the Pompeian camp that some of its members were already allocating priesthoods and dividing the expected spoils. While Pompeius considered defeating Caesar by starvation, his overconfident subordinates persuaded him to take the field. Pompeius also outnumbered Caesar and possessed the superior cavalry. Caesar's tactics led to the rout of cavalry, however, and in response, Pompeius fled, eventually being assassinated in Egypt.
Pharsalus was the turning point. Caesar was out numbered and could easily have lost but was the better general on the day. Remember that Pompey was considered the better general between the two at the time. After that, it was a matter of time before he finished the civil war. Pompey was later assassinated in Egypt and thus never had a chance to recover from the defeat at Pharsalus.
Here is a time line of events of the civil war between Caesar and Pompey (continued by his party after his death)
50 - Caesar enters Italy proper by crossing the Rubicon with his army
49 - Pompey evacuates Italy and arrives in Greece
48 - Caesar returns to Italy, crosses over to Greece, and barely wins the battle of Dyrrachium over Pompey
47 - After being reinforced by his ally Mithridates of Pergamum, Caesar, along with Cleopatra's forces, defeat her brother outside of Alexandria (Battle of the Nile), thereby ending the Egyptian phase of the war
46 - Caesar barely defeats his former lieutenant and current Pompeian, Titus Labienus at Ruspina in north Africa
45 - Caesar defeats the last pocket of Pompeian resistance at Munda in Spain. However, one of Pompey's sons, Sextus, escapes to Sicily where he builds up a base of power and resists Antony and Octavian until 35, when he was captured and killed in Miletus