Sanchez scored for Inter for only the second time but helped set up other goals as they consolidated third place in Serie A on 64 points and moved closer to a berth in next season's Champions League.
Brescia slipped to the bottom of the table on 18 points, eight adrift of the safety zone, and will need a dramatic turnaround in fortunes in their remaining nine games if they are to avoid an immediate return to Serie B.
"Sanchez is starting to do well," said Inter boss Antonio Conte.
"He's still not the Sanchez I liked in England, but he's on the right track. He did well against Parma, coming in with the right attitude. Today there was great commitment on his part, he played an excellent game."
Ashley Young gave Inter a fifth-minute lead before Sanchez, showing glimpses of old form, converted a 20th-minute penalty to double their advantage.
Sanchez netted a spotkick, after Moses had been bundled over, for his second Inter goal, nine months after his first against Sampdoria in late September.
D'Ambrosio started and finished the move for the third in the 45th minute, switching the ball wide to Young and then continuing his run to meet a cross to the back post, which he powerfully headed home.
Gagliardini's 53rd-minute goal came from a Sanchez free kick as he also stormed into the Brescia box to head home.
Inter kept up relentless pressure on their Lombardy rivals, bringing on both Romelu Lukaku and Eriksen for the last 20 minutes.
Lukaku's shot in the 83rd minute was spilled by Brescia goalkeeper Jesse Joronen and Eriksen followed up to scramble the ball over the line for his first league goal since moving from Tottenham Hotspur in January.
Substitute Candreva, who had earlier crashed an effort against the crossbar, finished the rout with a stinging shot two minutes from full-time.