Dobbs finished 8-for-15 for 100 yards with a touchdown pass and two interceptions.
Quarterback Josh Johnson, who started for the Giants with Eli Manning a healthy scratch, was unable to convert on two red-zone opportunities. New York's defense created the opportunities thanks to two turnovers recorded deep in Steelers' territory.
Johnson (5-for-10, 31 yards, three sacks) and the Giants ended up having to settle for field goals on both occasions, the first a 27-yarder by Aldrick Rosas and the second a 30-yarder by Mike Nugent to give New York a 6-0 lead.
The Steelers trimmed the Giants' lead thanks to a 48-yard field goal by kicker Chris Boswell midway through the second quarter, but the Giants responded on the next drive with a 52-yard Rosas field goal.
The Giants then committed the first of two turnovers that would lead to Steelers' scoring opportunities.
Punt returner Donte Deayon fumbled a punt, with the loose ball recovered by Steelers cornerback Mike Hilton. The miscue set up a 28-yard touchdown strike from Dobbs to receiver Cobi Hamilton.
Nugent gave the Giants the lead with his 45-yard field goal in the third quarer, but not for long thanks to the Steelers converting on another Giants turnover.
This time, linebacker Arthur Moats picked off Geno Smith's pass intended for receiver Travis Rudolph. That helped set the stage for what eventually became Boswell's 37-yard field goal to give his team a 13-12 lead.
The Steelers increased their lead to 20-12 with 12:56 left in the fourth quarter thanks to a 15-yard touchdown run by first-year running back Terrell Watson to cap a nine-play, 53-yard scoring drive spanning 4:45.
Smith was making his Giants debut and his first NFL appearance since suffering a season-ending ACL injury last year for the Jets. He finished 10-for-16 for 114 yards with one interception to lead the Giants quarterbacks.
Rookie quarterback Davis Webb, making his NFL debut for the Giants, finished 8-for-15 for 67 yards.
Orleans Darkwa finished as the leading rusher with 18 yards on three carries. Tight end Jerell Adams ended up as the Giants' leading receiver with three receptions for 28 yards.
For the Steelers, Hamilton finished as the leading receiver with two receptions for 72 yards and the touchdown. Running back Terrell Watson led the team in rushing with 10 carries for 41 yards and a score.