Robertson's deputy at left-back is now owned by a hefty 26 per cent of managers, making him the third most-popular defender in the game.
Tsimikas secured a clean sheet and six points at Norwich City on Saturday.
And as FPL managers continue to transfer him in, he has been among the first players to have a price rise.
With Robertson potentially missing until after the international break, the Greek can bring in more returns at home to Burnley in Gameweek 2.
Tsimikas was among five defenders priced at just £4.0m who started in Gameweek 1.
Brighton & Hove Albion's Shane Duffy had four shots in the penalty area against Burnley, more than any other player in his position in Fantasy.
But, like Tsimikas, both players have benefited from injuries to team-mates and are likely to be no more than short-term picks.
Ki-Jana Hoever and Tino Livramento were the other two sub-£4.5m defenders to start over the weekend.
If either can retain their roles, they could prove handy replacements for Tsimikas when Robertson returns.
Crucially, both have favourable schedules.
From Gameweek 4 onwards, each of Wolves’ subsequent four matches score just two in the Fixture Difficulty Ratings (FDR).
And following that period, Southampton’s four matches from Gameweek 9 onwards also score just two in the FDR.
Remember, the Gameweek 2 deadline is at 11:00 BST on Saturday 21 August.
Part 1: FDR Analysis: Spurs can thrive over short term
Part 2: GW1 FPL stats: Top captain Salah rivals Fernandes haul
Part 3: GW1 Bonus points: Antonio and Benrahma lead the way
Part 4: FPL Saturday review: Richarlison and Pogba in demand
Part 5: Fernandes and Salah to continue as top captains
Part 7: FPL experts: Are Son, Salah and Fernandes must-haves?