Dobie headed the opening goal midway through the second half and Davidson finished off the visitors with barely 10 minutes remaining, calming the nerves of the home faithful at McDiarmid Park.
Saints have gained a reputation as cup specialists in recent years, most notably during Owen Coyle’s spell as manager, but it is a tradition current boss Derek McInnes has been keen to maintain and they reached the semi-finals of this competition last season.
Goalkeeper Peter Enckelman (thigh) and midfielders Liam Craig (hamstring) and Jody Morris (hamstring) missed out due to injury for the hosts, while defender Dave Mackay failed to recover from a back problem.
Graeme Smith came in for Enckelman, while forwards Collin Samuel and Dobie both started, with Marcus Haber dropping to the bench following the weekend league defeat by Aberdeen.
Not surprisingly it was the home side who began the strongest, given the superior quality in their line-up, and former Scotland striker Dobie saw an early strike blocked by Stewart McCarthy.
When McCarthy fouled Sam Parkin on the right flank, it gave Saints another opportunity to put the ball in the Ton penalty area and Danny Grainger’s inswinging free-kick found Kevin Rutkiewicz.
Defender Rutkiewicz could not find a finish though, with his header causing Kevin Cuthbert no problems.
Morton demonstrated their menace just after the quarter-hour mark when Stewart Kean’s cross picked out Ryan Kane, whose header drew a save from Graeme Smith.
The goal which looked to be coming for Saints arrived in the 22nd minute when Samuel sent in a lofted centre from the left and Dobie headed in.
Saints had to face some Morton pressure after the goal, with Stewart Greacen unlucky when he put his shot just the wrong side of the post as Michael Duberry closed him down.
Cleveland Taylor was denied by Cuthbert at the other end, and Sam Parkin missed the target with a header, before Morton went close to equalising with half-time approaching when Smith turned Carlo Monti’s strike behind for a corner.
Morton boss Allan Moore replaced Kean with Peter Weatherson early in the second half, hoping the man who hit a double in the first-round rout of Stranraer could offer a greater threat than the player who scored a hat-trick in the same match.
Saints thought they should have had a penalty when Collin went down under the close attention of two Morton defenders, but their appeals were waved away, and soon afterwards Grainger’s free-kick had to be touched wide by Cuthbert.
Weatherson failed to hit the target after getting his head to Monti’s free-kick, shortly before Saints replaced Parkin and Samuel with Haber and Caddis.
Saints goalkeeper Smith denied McCarthy, pushing away his header, and Morton were finished off with 11 minutes remaining when Davidson struck.
Taylor looked favourite when he met Haber’s cross but he could not force the ball home and it was left to Davidson to tuck home the killer goal.