As a few other posters noted, there were many problems facing the Eastern Roman Empire in the sixth/seventh centuries, all of them combined meant that the western reconquests were either ephemeral or ended up costing the empire more resources than it gained. There were still some tangible gains though, as the wars brought the papacy under direct imperial control and there is no doubt that military victories bolstered imperial prestige, so the long-term cost-benefit ratio is debatable.
What I can say confidently however is that Justinian's reconquests did not save the Eastern Roman Empire - the empire before the conquest did not need saving, for the reigns of Anastasius and Justin I were periods of imperial reconsolidation in terms of finance and infrastructure projects. The Vandals and the Ostrogoths were relatively weak in the 520s/530s, hence Justinian pounced from on them from a position of strength, not because his empire was in decline. Even after the reconquest the west contributed very little to the centre, as the long wars proved devastating to local economies. In any case, the empire did not need saving in the late sixth century either, as we have records of the emperors Tiberius and Maurice occasionally sending aid to the west to deal with the Lombards, rather than anything significant being sent east. Admittedly North Africa was still a wealthy province, but we have little to no evidence of what they actually contributed to the still vigorous empire. Italy was a non-issue at this stage, since it was still recovering from the Gothic War and was now contested between the Romans and the Lombards. There are even fewer records for Roman Spain, except in terms of archaeology, which unfortunately doesn't tell us much about how the small Roman enclaves affected the empire as a whole.
The seventh century was however an extended crisis for the empire, but problems only really began in 608, when the rebellion of Heraclius in North Africa caused a civil war that led to the collapse of imperial defences against the Persians, which is hardly a shining example of the western provinces saving the east. From then on there is even less of a case for the west saving the east, since western exarchs repeatedly rebelled and had to be put down - on one occasion an exarch of Africa was killed by Arab raiders rather than a Roman expedition, demonstrating precisely how little control the Eastern Romans had over their western holdings. Certainly, it was this weakness that contributed to Constans II's decision to move to Italy in the 660s, most likely to reassert imperial control after two major rebellions (possibly even usurpations) and to secure western defences against Arab naval raids. Again, it was evidently not a case of western resources being particularly useful to the empire. With Constans II's murder (with a bucket) in 668, the imperial centre once again moved back to Constantinople and from this point onwards the west became less and less important to an increasingly embattled empire, even though the empire still controlled parts of Italy until the eleventh century.
The main argument between historians is whether they ultimately weakened the ERE, and contributed to the loss of Egypt and the Middle East to the Muslims after 634. But I suppose there is always an angle.
Probably the one significant contribution from the area of Justinian's conquests was that Heraclius (emperor 610-641), used the troops from Africa to stage a coup against Phocas, who had suffered reverses during the second Byzantine-Sassanid war. That said, the war would continue for another 17 years and include occupation of most of the empire by the Persians and the Avars, and a siege of Constantinople before it finally ended with the Sassanids returning all conquered lands in 627. Of course, soon after, the Arabs started their conquests and the ERE would lose the levant, Egypt and Africa permanently.
Are you perhaps begging a question? I think it'd be better to ask "Did Justinian I's military conquests save the Eastern Roman Empire?" I am no expert -- I wish I could answer the questions -- but I believe there's a strong opinion that the answer is "no, they hurt":