Robinson's lightning reflexes helped him push a blistering Fabricio Coloccini drive over the bar in the 54th minute as the visitors dominated after the break.
Roque Santa Cruz had headed a ninth minute effort against the post for Rovers, who toiled for much of the afternoon while Peter Lovenkrands lofted a 14th minute effort onto the top of the bar.
Magpies boss Alan Pardew had been able to name an unchanged side following the stunning comeback against Arsenal last week, with stop-gap striker and former Rovers star Shefki Kuqi on the bench.
With Rovers having come off worse in a seven-goal thriller at Wigan, it was not surprising both teams started where they left off and produced a richly entertaining first quarter-hour.
Jermaine Jones' attempt at a Wayne Rooney-style overhead kick in the third minute failed to have quite the same impact as the American missed the ball and clattered Mike Williamson.
Newcastle came close two minutes later when Lovenkrands played in Jose Enrique down the left and his ball across the face of goal was palmed clear by Paul Robinson.
In what was warming into a feisty encounter, Cheik Tiote was warned for a rash challenge on Nikola Kalinic, and Kevin Nolan and David Dunn almost came to blows on the half-way line.
Rovers should have grabbed the lead in the ninth minute when captain Christopher Samba's flick in the box set up Santa Cruz for a header which struck Steve Harper's right-hand post.
Lovenkrands had an even better chance for the visitors on 14 minutes when he was put clear by Enrique but his left-foot lob over Robinson dropped onto the top of the bar.
As the fast pace continued Rovers had another chance when Dunn blazed over the bar after good work from Junior Hoilett down the Rovers left.
Michel Salgado gifted Newcastle an opening in the 25th minute with a wayward cross-field pass which let in Lovenkrands, but the Dane ran into Ryan Nelsen who cleared.
Nelsen headed away another cross from the dangerous Enrique as the game's early excitement faded with a series of minor infringements ruining any momentum the game could gather.
The visitors dominated from the start of the second period against a Blackburn side forced to make two changes due to injuries to Santa Cruz and Junior Hoilett.
Joey Barton had the first chance of the second half for Newcastle after Jonas Gutierrez's charge down the left, but Barton's swept left-foot effort sailed well over the bar.
Newcastle came closer on 54 when Coloccini chested down a Barton corner from the left and rammed a rising left-foot shot which Robinson did superbly to tip over the bar.
Barton almost found Best in the box on the end of a neat one-two and as Rovers struggled to clear it was Gutierrez's turn to fire the loose ball over the bar.
Rovers were being pinned back in their own half with captain Samba outstanding. But without the presence of Santa Cruz they were struggling to create anything in the visitors' half.
After completely dominating for the first half hour of the second half, Newcastle lived on their nerves in the 81st minute when a deep Pedersen corner caused chaos in the box.
If the near-miss was designed to belatedly stir the home side into life it failed to work as both Barton and Gutierrez were only let down at the other end by the quality of their final balls.
Pardew threw Kuqi into the fray for the three minutes of added time no doubt in the hope of a fairytale start in the black and white stripes but referee Stuart Atwell had other ideas.