A nine-minute spell before half-time – during which Rodrigo de Paul was shown a red card for striking Antonio Candreva on the head following the intervention of VAR and Sensi nodded Inter ahead – proved decisive.
Romelu Lukaku endured a disappointing outing and made way for Lautaro Martinez in the 65th minute, before Sanchez replaced Matteo Politano, the Chile forward making his first appearance for the Nerazzurri against one of his former clubs.
While Inter were unable add to their tally, they had already done enough to pick up a third straight win under Antonio Conte and move two points clear of Juventus at the top of the table.
Politano went close to putting Inter ahead in the fourth minute when his low drive from 25 yards struck the outside of the post.
Sensi saw a fine volley from the edge of the box pushed aside by Udinese goalkeeper Juan Musso, before Walace was kept out by Samir Handanovic at the other end.
Udinese were reduced to 10 men after 35 minutes when De Paul was shown a red card after referee Maurizio Mariani reviewed footage of his retaliation to Candreva appearing to touch his face.
Inter took advantage a minute before half-time, Sensi getting across Rodrigo Becao and heading a fine delivery from Diego Godin – making his first start since joining from Atletico Madrid – in off the underside of the crossbar.
Kevin Lasagna overpowered Godin but was thwarted by Handanovic, before Musso denied Politano and Roberto Gagliardini – a half-time replacement for Nicolo Barella – before the hour mark.
After Musso kept out a Sensi free-kick, an awkward fall for Politano paved the way for Sanchez to enter the fray to a great ovation with 10 minutes remaining.
Sanchez went close to marking his debut with a goal when Candreva’s square pass fell to him in the six-yard box, but a scrambling Musso kept him at bay and Inter coasted to the final whistle.