Searching for consecutive wins for only the second time all season, the Warriors looked set to achieve their aim after bossing much of the contest at Toyota Stadium.
A determined defence saw the Warriors hold an 11-point lead with 10 minutes to go, but that same defence then went to pieces as Sharks halfback Robson crossed twice in four minutes to steal victory.
The loss was the Warriors' eighth in 14 matches this season and dealt a blow to their finals hopes, but last year's beaten grand finalists are still only four points outside the top eight heading into next week's bye.
Omar Slaimankhel was called into the side as a late replacement for Manu Vatuvei who was ruled out with a knee injury, and the former Afghani refugee's debut got off to an inauspicious start.
Slaimankhel has scored 45 tries in 45 Toyota Cup appearances for the Warriors' under-20 team, but his immediate impact was felt at the other end of the pitch. With his third touch Slaimankhel lost control while hitting up the ball near the Warriors' line, and from the ensuing possession Ben Pomeroy put the Sharks in front.
Their lead didn't last as the Warriors hit back in familiar fashion. Shaun Johnson's boot was the creator of the Warriors' first two tries during their win over Penrith on Monday night, and this time his grubber was barely grounded at the back of the in-goal by Kevin Locke to level the scores.
And provider then turned scorer, when Johnson opportunistically grabbed a James Maloney bomb to put the Warriors in front.
Maloney's conversion saw him become just the second man after Stacey Jones to score 500 points for the club, and he celebrated with a drop goal before David Carney's penalty right on halftime reduced the Warriors' lead to 13-8.
The Warriors spent the majority of the first ten minutes in the second spell camped in the Cronulla half, and they soon converted their territorial dominance into an extended advantage on the scoreboard.
Konrad Hurrell looked to have blown a golden chance when he sent a pass behind the unmarked Bill Tupou and over the sideline, but he immediately made amends by muscling his way over close to the line for his seventh try in five matches.
Maloney then provided the Warriors with the rarest of things - their first penalty goal in 14 matches this season - to increase the lead to 19-8.
With 10 minutes remaining it looked like the Warriors were set to get back-to-back wins for only the second time this season but Robson struck a double blow to put Cronulla in the lead.
First, Carney put the halfback through a hole and a protracted slide on the rain-sodden turf took him over the line, then Carney's 40/20 saw the Sharks regain the ball before Robson's jinking run edged them in front.
The visitors threatened to pull off a last-gasp win but a goalline stand from the Cronulla defence consigned the Warriors to their eighth loss.
Sharks 20 (Pomeroy, 2 Robson tries; Carney pen, 3 cons)
Warriors 19 (Locke, Hurrell tries; Maloney pen, con, dg)
HT: 8-13
