Steve McClaren's men took just ten minutes to move ahead as Hasselbaink rounded off a fine passing move to notch his first goal of the season.
They then had to battle to hold off a spirited second-half fightback from the Swiss side to secure a victory that already leaves them well placed to qualify for the knockout stages.
Hasselbaink was recalled to the starting line-up along with Mark Viduka and Szilard Nemeth, with Yakubu Ayegbeni, Massimo Maccarone and Franck Queudrue stepping down from the team that laboured to a 1-1 draw with Portsmouth last weekend.
Boro took control from the kick-off and Gaizka Mendieta forced home keeper Fabio Coltorti into a save with a first-minute free-kick after Viduka was fouled by ex-Arsenal defender Igor Stepanovs.
Coltorti used his legs to deny Viduka in the seventh minute but was powerless to prevent the visitors taking the lead three minutes later.
Nemeth and Doriva created the space for Hasselbaink and the Dutchman applied a cool side-footed finish to break the deadlock.
Middlesbrough should have doubled their lead six minutes later after Hasselbaink and Mendieta set Nemeth free, but he inexplicably looked to pass instead of taking advantage of a golden shooting opportunity.
Visiting keeper Mark Schwarzer had a lucky escape in the 21st minute when he spilled a long-range shot from Rogerio and Eduardo fired home the rebound. However, Grasshoppers' celebrations were cut short by an offside flag.
The hosts should have levelled after a lively start to the second half, but Eduardo somehow headed over from Scott Sutter's cross.
Boro almost immediately punished Eduardo's error but Doriva dragged his shot wide when a simple pull back would have presented Viduka with a tap-in.
Hasselbaink then saw an effort chalked off by an assistant referee's flag after an exquisite pass from Mendieta set him free.
However, it was the hosts who controlled the closing stages and Boro needed centre-back pairing Gareth Southgate and Chris Riggott to be at their best as they grimly clung on to their lead.
They were also thankful to the both the referee and Schwarzer.
The official waved away penalty appeals after Ricardo Cabanas headed to the turf in the Boro box and later when Doriva appeared to handle a Cabanas cross.
The Australian shot-stopper had to dive full length to tip Rogerio's header to safety and also made a vital interception from Sutter's cross to ensure Boro clinched an important victory.