Fly-half Chris Noakes scored all of the hosts' points at North Harbour Stadium with six straight penalty kicks but the match was nearly stolen by an inspirational two-try effort by Stormers captain Jean de Villiers.
With the visitors trailing 15-3 in the final quarter, the bulky inside centre willed his team back into the contest with a 65th minute try and another with three minutes left on the clock.
Full-back Joe Pietersen's second conversion made it a one-point ball game, but the Blues were composure personified in a frantic four minutes after the siren, withstanding a more-than-20 phase assault in their territory as the Stormers searched in vain for a winning penalty kick.
It was the South Africans who ultimately crumbled under the pressure, as Pietersen miscued a speculative drop goal attempt from long range to surrender the match.
The win was the Blues' fifth from nine games midway through the season and keeps the pressure on the defending champion Chiefs in the New Zealand conference.
The Stormers were left to rue a sloppy first half in which indiscipline cost them a 9-3 lead at the break and sloppy kicking cost them a number of scoring chances.
After Noakes slotted two penalty goals in the opening quarter, Stormers lock Andries Bekker failed to plant the ball over the line at the end of a promising drive in the 19th minute.
Barring a penalty goal from point-blank range, Pietersen's normally dependable boot deserted him, and the Stormers waved away two chances to kick for points.
They came away empty-handed both times after being frustrated by a dogged Blues defence and a Pietersen kick that failed to clear the touch-line.
In between, Pietersen missed a penalty goal from straight in front and watched ruefully as Noakes kicked his third six minutes before halftime.
The fly-half made it five-from-five with another two penalty goals early after the break, and the Stormers appeared destined to slump to an insipid loss before De Villiers's inspired finish.
De Villiers offloaded to fleet-heeled outside centre Juan de Jongh who was brought down just short of the line, but the skipper collected the ball back and barged through a clutch of defenders for his first try.
He ratcheted up the pressure again by shrugging off a Francis Saili tackle and bolting 40 metres to the line to score his second, but the Blues held on grimly to secure the points in a frenetic atmosphere.