Warnock named eight debutants, including Varney and Norris, and teenage defender Sam Byram, with the hosts taking the lead through Becchio in the 20th minute.
Another summer signing, Rodolph Austin, fired in a shot from outside the penalty area and, although Shrewsbury goalkeeper Chris Weale got down well to keep the ball out, Argentinian Becchio slid home the rebound from six yards.
Leeds doubled their advantage six minutes later following a sweeping move that ended with striker McCormack's shot blocked, allowing Varney to pounce and apply a neat finish.
Shrewsbury's Luke Summerfield lashed a shot wide for the visitors soon after, while team-mates Paul Parry and Jermaine Grandison had both gone close with long-distance efforts earlier.
But Leeds looked the more assured in possession and midfielder Norris twice went close to adding a third, before eventually slamming home a shot into the bottom corner from Varney's headed knockdown.
McCormack stroked his penalty into the bottom corner to make it 4-0 in the 70th minute after Shrews defender Michael Hector handled in the area.
Mathias Pogba marked his debut with a double as newly-promoted Crewe dumped Hartlepool out of the Capital One Cup with a crushing 5-0 victory.
A first-half strike by winger Matty Robson enabled Carlisle to see off a determined challenge from League Two side Accrington Stanley as the Cumbrians held out for a 1-0 win.
Ajay Leitch-Smith (seven) and Mathias Pogba (31) werealso on target for the Railwaymen before half-time while Pogba added another in the final minute to make it five.
Middlesbrough edged into the next round with a 2-1 scoreline against Bury at Gigg Lane, with Marvin Emnes (32) and Grant Leadbitter (61) securing Boro's passage.
Chris Dagnall spared team-mate Ben Alnwick's blushes by firing Barnsley to a 4-3 win over Rochdale after extra-time at Spotland.
Stale Solbakken started life as Wolves boss with an unconvincing victory after his side just about edged past Aldershot on penalties in their Capital One Cup first-round clash at Molineux.
Solbakken's side, who are bidding to secure an immediate return to the Premier League, were desperately short of quality against a plucky Shots side who pushed them all the way.
Sylvan Ebanks-Blake's well-taken finish opened the scoring in the second half before the League Two visitors, who made the fourth round last term before bowing out to Manchester United, quickly levelled matters through Michael Rankine's header.
And after both sides were put through a draining 120 minutes on the opening day, it was left for Wanderers keeper Carl Ikeme to make the decisive shoot-out save to deny Jordan Roberts from the spot and seal a narrow 7-6 triumph.
Steve Bruce began his reign as Hull manager in dramatic fashion as the Tigers progressed to the Capital One Cup second round with a 7-6 penalty shoot-out victory over Rotherham.
The Championship side had to come from behind after summer signing Lionel Ainsworth made an immediate impact from the bench, putting the Millers ahead in the 52nd minute with an unstoppable volley from the edge of the box.
Bruce's own super sub Aaron Mclean spared his blushes with an equaliser after 70 minutes.
Summer signing Ashley Hemmings scored on his debut to send Walsall into the second round of the Capital One Cup with a 1-0 win at home to 10-man Brentford.
Substitute Chris Iwelumo's extra-time goal secured a narrow 1-0 victory over Wycombe in Gianfranco Zola's first competitive game in charge of Watford.
Burton Albion pulled off a dramatic victory at Brammall Lane as they beat Sheffield United 5-4 on penalties after drawing 2-2.
Notts County also lost at home, losing 1-0 to Bradford City after extra time.
And Rochdale also missed out after extra time on home turf, losing 4-3 to Barnsley at Spotland.
And finally, Doncaster beat local rivals York 4-2 on penalties after a 1-1 draw at the Keepmoat Stadium.