Nathan Doak contributed three conversions and Jamie Benson two in six tries on the afternoon for Richie Murphy’s side.
Hooker Walker, who made his Ulster debut two weeks ago, made his first competitive start and opened the scoring in the 12th minute.
Last season Challenge Cup The finalists raced to a 19-0 advantage at half-time, thanks to two Walker tries, with Jacob Stockdale scoring eight minutes before and one after.
Giovanni Licata put the hosts on the scoresheet with a try three minutes after the break, converted by Giacomo Da Re, but Dave McCann replied four minutes later.
Alessandro Ortombina and Giovanni Montemauri – who also added two conversions – began to claw at Ulster’s lead, but tries from Michael Lowry and Martin Moloney in the final 10 minutes sealed victory.
meanwhile, Scarlets ran in seven tries in a comprehensive 47-14 Challenge Cup win over Newcastle.
The home team led 26-0 at halftime and did not let up in the one-sided start to the campaign.
Scarlets’ first try came in the 12th minute through Sam Costelow, who added the conversion, with Fletcher Anderson coming over nine minutes later.
Ex Newcastle center Johnny Williams was the next to go in and Gareth Davies added a fourth try just before half-time.
After passing forward, Scarlets had a fifth try disallowed two minutes into the second half, while Newcastle’s appeals for Obi Ene’s try were also dismissed by the video referee, and the away side were then reduced to 14 men when Murray McCallum was sent to the sin bin.
Ellis Mee and a Macs Page double extended the Scarlets’ lead, although Newcastle managed a spirited finish when Oli Spencer and Ollie Leatherbarrow went over.