Trailing following a contentious 2-1 defeat in Bilbao last week, Barca had to wait for their breakthrough before it arrived in landmark fashion 10 minutes from half-time.
Luis Suarez volleyed his 100th goal for the club on the end of a move that fittingly featured Messi and Neymar – it was the 300th goal for MSN since the awesome forward trio began playing together in 2014.
Neymar won and converted a penalty early in the second period, but Athletic hit back through wideman Enric Saborit, a Barcelona-born product of the Espanyol youth system.
But Barcelona, looking for a win at their third attempt in 2017, were not discouraged and a familiar hero stepped forward.
Having arrowed a set-piece into the top left corner to snatch a 1-1 draw at Villarreal on Sunday, Messi deftly whipped an effort from similar range across the Bilbao wall and in off the right-hand post to stylishly book a quarter-final spot.
The first leg at San Mames prompted Gerard Pique's week-long critique of Spanish officials and controversy arrived again after 26 keenly fought minutes, when Barcelona briefly though they were level on aggregate.
Athletic were forced into a change before the half-hour mark, with defender Xabier Etxeita coming off worse after hacking down Neymar.
Presented with a free-kick opportunity in a similar position to his last-gasp equaliser versus Villarreal on Sunday, Messi curled an attempt over, with Neymar having come in for more robust attention.
The visitors' concerns over the damage Neymar might cause proved well-founded in the 35th minute when he collected Messi's pass and floated a cross from the left channel for Suarez to gleefully smash home on the volley and bring up his landmark.
MSN's collective triple century became 301 three minutes into the second half, with the latest foul on Neymar coming when Athletic right-back Eneko Boveda went to ground as his man burst into the left-hand side of the area.
The 24-year-old picked himself up and, following a prolonged stuttering run-up, he sent Gorka Iraizoz the wrong way.
But any sense of Barca having completed a turnaround proved short-lived when first-leg goalscorer Inaki Williams – passed fit to play despite a bout of illness - crossed from the right for Saborit to head back across goal and beyond Marc-Andre ter Stegen.
Suarez stabbed an effort the wrong side of the post after Messi sublimely scooted around Aymeric Laporte, while left-back Jordi Alba surged on to a trademark Iniesta pass but shot too close to Iraizoz.
Alba's give-and-go with Neymar presented with Suarez with a 68th-minute chance that the usually masterful marksman slashed wide.
Messi did not lack for precision when the Uruguay striker was felled by Benat Etxebarria 13 minutes from time, sending Camp Nou into raptures.
Nevertheless, the focus at full-time rightly remained on Barca's mercurial front three at full-time - 302 not out.