Barhey fired 102 off 128 deliveries, with eight fours and three sixes, and Landsdale struck 91 off 75 balls for the third wicket as the hosts, put into bat, rattled up 265-4 in their 50 overs.
Attock’s reply was positive but they fell short by 23 runs.
Though Charlie Robards fell for eight and he was soon followed by Matthew Ewer (38) to leave Rugby 88-2, Barhey and Landsdale set about the Attock bowlers with renewed vigour to power the score to 250. Qasim Shah was the pick of the bowlers with 3-34.
Attock lost their first wicket with just three runs on the board,. Ewer dismissing Mohammed Ishtiaq, but soon recovered with Arsalaan Mahmood (70) and Daniyal Khan (51) combining in a 125-run stand.
Once they were separated Asim Khan (37) kept up the good work and Attock closed down their target.
But with last man Qasim unable to bat, the departure of Adam Mahmood for six off the first ball of the 48th over signalled the end of their hopes.
Rugby, who lie fifth, head to Moseley Ashfield on Saturday.
Willoughby head into a busy double header weekend of league games and the prospect of a challenging August on the back of victory at bottom-of-the-table Podington Seconds in Northamptonshire League Division Six.
Having kept the hosts to a par score of 155-9 off their allotted 45 overs, Mark Fell taking 4-20, the visitors were in early trouble being reduced to 22-4.
However, they recovered and eventually reached their target total with six overs and four wickets in hand. Ashley Rayner top-scored with 63.
Willoughby entertain Stony Stratford 3rds on Saturday and then welcome Rothwell Town 2nds on Sunday.