The former Norwich striker, who spent much of last season on the bench at Leyton Orient, fired the Saddlers ahead from a disputed 15th-minute penalty at Dean Court on Saturday and doubled the lead early in the second half with a header.
Walsall rode their luck early on, with Darryl Westlake’s desperate goal-line clearance denying Shaun MacDonald a goal on his Bournemouth debut.
Minutes later Liam Feeney rolled a shot inches wide for the Cherries.
Walsall snatched a controversial lead with their first attack in the 15th minute. Feeney was penalised – and booked – for handling a cross from Mat Sadler and Jarvis drilled the resulting spot-kick into the bottom corner for his first Walsall goal.
Saddlers goalkeeper David Grof saved from Adam Barrett and Harry Arter and Jarvis secured the points nine minutes after the break when he headed in Kevan Hurst’s corner.