In a game of few chances, Jason Roberts came closest to breaking the deadlock just seconds after Stephen Warnock had struck the bar midway through the second half.
Meanwhile, Everton remain rooted in sixth place after Steven Pienaar saw a shot spectacularly saved and Joleon Lescott fired inches over in the 90th minute.
Before the game, Rovers boss Sam Allardyce was forced to make two changes to the side that defeated Hull City three days ago.
Goalkeeper Jason Brown was handed just his just third league start in a Rovers shirt in place of the injured Paul Robinson, while Roberts replaced the suspended Morten Gamst Pedersen.
With one eye on Sunday's FA Cup quarter-final against Middlesbrough, Everton made three changes to the side that beat West Brom at the weekend, with Lescott, Jack Rodwell and Leon Osman coming in for Tony Hibbert, Dan Gosling and Marouane Fellaini.
In bitterly cold conditions, both sides made a bright start to the proceedings with Jo firing a deflected volley inches wide of the post with just seconds on the clock.
Rovers responded well with Warnock and El Hadji Diouf both seeing crisp strikes blocked in quick succession and ten minutes later, Keith Andrews took too long to get a left-foot shot away on the edge of the area after good work by Roque Santa Cruz down the left.
A bizarre passage of play continued with both sides persistently guilty of giving away possession and the home fans sarcastically applauding every decision that went in their favour from referee Alan Wiley.
The best chance of the half arrived on 39 minutes when Phil Jagielka centred from the right and Pienaar smashed the ball goalwards, but Brown produced a stunning reaction save to keep the shot out.
Minutes later, another excellent Pienaar effort struck the arm of Andre Ooijer inside the box but the Dutchman knew little about it and the half-hearted penalty appeals were ignored.
Tugay replaced Vince Grella at the break as the home side looked for an early breakthrough.
On 50 minutes, Warnock laid off Roberts' right-wing cross to the fast-arriving Santa Cruz, who fired agonisingly wide of the far post.
Headers from Tim Cahill and Santa Cruz came and went before Rovers had another golden opportunity to take the lead.
On 67 minutes, Warnock's free-kick bounced back off the bar and just 30 seconds later, Roberts ran on to Aaron Mokoena's throughball but, after out-pacing the Everton defence, he failed to beat Tim Howard.
As the contest drew to a close, Brown was well placed to keep out Cahill's lethal long-range strike and in the final minute, the Australia international flicked on an Everton corner from the left, but Lescott volleyed narrowly over the bar.